Gram matrices and Stirling numbers of a class of diagram algebras, II
In the paper [6], we introduced Gram matrices for the signed partition algebras, the algebra of Z₂-relations and the partition algebras. (s₁, s₂, r₁, r₂, p₁, p₂)-Stirling numbers of the second kind are also introduced and their identities are established. In this paper, we prove that the Gram matrix...
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| Cite this: | Gram matrices and Stirling numbers of a class of diagram algebras, II / N. Karimilla Bi, M. Parvathi // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 215–256. — Бібліогр.: 18 назв. — англ. |
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Karimilla Bi, N. Parvathi, M. 2023-02-25T14:47:49Z 2023-02-25T14:47:49Z 2018 Gram matrices and Stirling numbers of a class of diagram algebras, II / N. Karimilla Bi, M. Parvathi // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 215–256. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC: 16Z05 https://nasplib.isofts.kiev.ua/handle/123456789/188360 In the paper [6], we introduced Gram matrices for the signed partition algebras, the algebra of Z₂-relations and the partition algebras. (s₁, s₂, r₁, r₂, p₁, p₂)-Stirling numbers of the second kind are also introduced and their identities are established. In this paper, we prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. As a consequence, the semisimplicity of a signed partition algebra is established. The authors would like to express their gratitude and sincere thanks to the referee for all his(her) valuable comments and suggestions which in turn made the paper easy to read. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Gram matrices and Stirling numbers of a class of diagram algebras, II Article published earlier |
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Gram matrices and Stirling numbers of a class of diagram algebras, II |
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Gram matrices and Stirling numbers of a class of diagram algebras, II Karimilla Bi, N. Parvathi, M. |
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Gram matrices and Stirling numbers of a class of diagram algebras, II |
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Gram matrices and Stirling numbers of a class of diagram algebras, II |
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Gram matrices and Stirling numbers of a class of diagram algebras, II |
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Gram matrices and Stirling numbers of a class of diagram algebras, II |
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gram matrices and stirling numbers of a class of diagram algebras, ii |
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In the paper [6], we introduced Gram matrices for the signed partition algebras, the algebra of Z₂-relations and the partition algebras. (s₁, s₂, r₁, r₂, p₁, p₂)-Stirling numbers of the second kind are also introduced and their identities are established. In this paper, we prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. As a consequence, the semisimplicity of a signed partition algebra is established.
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Gram matrices and Stirling numbers of a class of diagram algebras, II / N. Karimilla Bi, M. Parvathi // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 215–256. — Бібліогр.: 18 назв. — англ. |
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AT karimillabin grammatricesandstirlingnumbersofaclassofdiagramalgebrasii AT parvathim grammatricesandstirlingnumbersofaclassofdiagramalgebrasii |
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2025-11-26T01:45:40Z |
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1850606141392814080 |
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“adm-n2” — 2018/7/24 — 22:32 — page 215 — #53
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 2, pp. 215–256
c© Journal “Algebra and Discrete Mathematics”
Gram matrices and Stirling numbers
of a class of diagram algebras, II
N. Karimilla Bi and M. Parvathi
Communicated by R. Wisbauer
Abstract. In the paper [6], we introduced Gram matrices
for the signed partition algebras, the algebra of Z2-relations and
the partition algebras. (s1, s2, r1, r2, p1, p2)-Stirling numbers of the
second kind are also introduced and their identities are established.
In this paper, we prove that the Gram matrix is similar to a matrix
which is a direct sum of block submatrices. As a consequence, the
semisimplicity of a signed partition algebra is established.
1. Introduction
In this paper, we establish that the Gram matrices Gk
2s1+s2
and
−→
Gk
2s1+s2
introduced in [6] are similar to matrices G̃k
2s1+s2
and
−̃→
G
k
2s1+s2
respectively
and each of which is a direct sum of block sub matrices Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2 of sizes f2r1+r2
2s1+s2
and
−→
f 2r1+r2
2s1+s2
respectively. The diagonal
entries of the matrices Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2 are the same
and the diagonal element is a product of r1 quadratic polynomials and
r2 linear polynomials which could help in determining the roots of the
determinant of the Gram matrix. Similarly, we have also established that
the Gram matrix Gk
s of a partition algebra is similar to a matrix G̃k
s which
is a direct sum of block matrices Ãr,r of size f r
s . The diagonal entries of
2010 MSC: 16Z05.
Key words and phrases: Gram matrices, partition algebras, signed partition
algebras and the algebra of Z2-relations.
“adm-n2” — 2018/7/24 — 22:32 — page 216 — #54
216 Gram matrices and Stirling numbers
the matrices Ãr,r are the same and the diagonal element is a product of
r linear polynomials which could help in determining the roots of the
determinant of the Gram matrix.
Using the cellularity structure defined in [5], we show that the algebra
of Z2-relations and signed partition algebras are semisimple over K(x)
where K is a field of characteristic zero and x is an indeterminate and
it is also semisimple over a field of characteristic zero except for a finite
number of algebraic elements and we also prove that the algebra of Z2-
relations and the signed partition algebras are quasi-hereditary over a
field of characteristic zero. In particular, if q is an integer 6 k − 2 and q
is a root of the polynomial x2 − x− 2r′, 0 6 r′ 6 k − 2 then the algebras
AZ2
k (q) and
−→
AZ2
k (q) are not semisimple.
2. Stirling numbers of second kind of the algebra of
Z2-relations, signed partition algebras and partition
algebras
Lemma 2.1.
(a) In the algebra of Z2-relations, let dp1,p2i,α , d
r1,r2
j,β ∈ J2k
2s1+s2
with 2p1 +
p2 < 2r1 + r2 then d
p1,p2
i,α is coarser than d
r1,r2
j,β if and only if
l(dp1,p2i,α .d
r1,r2
j,β ) = 2p1 + p2 where J2k
2s1+s2
is as in Notation 3.6(a)
in [6].
(b) In signed partition algebras, let d
p1,p2
i,α , d
r1,r2
j,β ∈
−→
J 2k
2s1+s2
with 2p1 +
p2 < 2r1 + r2 then d
p1,p2
i,α is coarser than d
r1,r2
j,β if and only if
l(dp1,p2i,α .d
r1,r2
j,β ) = 2p1 + p2 where
−→
J 2k
2s1+s2
is as in Notation 3.6 (b)
in [6].
(c) In partition algebras, let Rd
p
i,α , R
drj,β ∈ Jk
s with p < r then Rd
p
i,α is
coarser than R
drj,β if and only if l(Rd
p
i,α .R
drj,β ) = p where Jk
s is as in
Notation 3.6(c) in [6].
Proof. Part (a): d
p1,p2
i,α is coarser than d
r1,r2
j,β if and only if every {e}-
through class of dr1,r2j,β is contained in a {e}-through class of dp1,p2i,α , every
Z2-through class of dr1,r2j,β is contained in a Z2-through class of dp1,p2i,α , every
{e}-horizontal edge of dr1,r2j,β is contained in either a {e} or Z2-horizontal
edge or {e}-through class of dp1,p2i,α and every Z2-horizontal edge of dr1,r2j,β
is contained in a Z2-horizontal edge or Z2-through class of dp1,p2i,α .
Thus, the number of loops in the product d
p1,p2
i,α .d
r1,r2
j,β is 2p1 + p2.
The proof of (b) and (c) are similar to the proof of (a).
“adm-n2” — 2018/7/24 — 22:32 — page 217 — #55
N. Karimilla Bi, M. Parvathi 217
Lemma 2.2 ([18]). Given any two diagrams d
r1,r2
i,α and d
r′
1
,r′
2
j,β such that
♯p(dr1,r2i,α .d
r′
1
,r′
2
j,β ) = 2s1 + s2 then there exists a unique diagram which is the
smallest diagram d
r′′
1
,r′′
2
l,γ among the diagrams coarser than both d
r1,r2
i,α and
d
r′
1
,r′
2
j,β .
Also, l(d
r′′
1
,r′′
2
l,γ .d
r′′
1
,r′′
2
l,γ ) = l(d
r′′
1
,r′′
2
l,γ .d
r1,r2
i,α ) = l(d
r′′
1
,r′′
2
l,γ .d
r′
1
,r′
2
j,β ).
Proof. The proof follows from Definition 2.13 in [6]. and [18].
2.1. Column operations on the Gram matrices of the algebra
of Z2-relations, signed partition algebras and partition al-
gebras
We now perform the column operations inductively on the Gram
matrices of the algebra of Z2-relations, signed partition algebras and
partition algebras as follows:
Let d
0,0
i,α be coarser than d
0,1
j,β . Then by Lemma 2.2,
l(d0,0i,α.d
0,0
i,α) = l(d0,0i,α.d
0,1
j,β) = 0.
We apply the column operation: L(j,β,0,1) → L(j,β,0,1) − L(i,α,0,0) then
the ((i, α, 0, 0), (j, β, 0, 1))-entry becomes
a(i,α,0,0),(j,β,0,1) − a(i,α,0,0),(i,α,0,0) = 1− 1 = 0.
Similarly, apply the column operations L(j,β,r′
1
,r′
2
) → L(j,β,r′
1
,r′
2
) −
L(i,α,r1,r2) whenever d
r1,r2
i,α is coarser than d
r′
1
,r′
2
j,β .
Then b(i,α,r1,r2),(j,β,r′1,r
′
2
) denotes the ((i, α, r1, r2), (j, β, r
′
1, r
′
2))-entry
after all the column operations are carried out
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = a(i,α,r1,r2),(j,β,r′1,r
′
2
)
−
∑
d
r′′′
1
,r′′′
2
l,γ
>d
r1,r2
i,α
d
r′′′
1
,r′′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(i,α,r1,r2),(l,γ,r′′′1
,r′′′
2
)
−
∑
d
r′′′′
1
,r′′′′
2
k′,δ′
>d
r′
1
,r′
2
j,β
d
r′′′′
1
,r′′′′
2
k′,δ′
6>d
r1,r2
i,α
b(i,α,r1,r2),(k′,δ′,r′′′′1
,r′′′′
2
)
(2.1)
Lemma 2.3. (a) In the algebra of Z2-relations and signed partition alge-
bras, let (i, α, r1, r2) < (j, β, r′1, r
′
2).
“adm-n2” — 2018/7/24 — 22:32 — page 218 — #56
218 Gram matrices and Stirling numbers
(i) If dr1,r2i,α is coarser than d
r′
1
,r′
2
j,β then
b(j,β,r′
1
,r′
2
),(i,α,r1,r2) = b(i,α,r1,r2),(i,α,r1,r2).
(ii) If dr1,r2i,α is not coarser than d
r′
1
,r′
2
j,β and l(dr1,r2i,α .d
r′
1
,r′
2
j,β ) > 0 then
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = 0 and b(j,β,r′
1
,r′
2
),(i,α,r1,r2) = 0.
(iii) If dr1,r2i,α is coarser than d
r′
1
,r′
2
j,β then
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = 0
where b(i,α,r1,r2),(j,β,r′1,r
′
2
) is the ((i, α, r1, r2), (j, β, r
′
1, r
′
2))-th entry
after all the column operations are carried out.
(b) In partition algebras, let (i, α, r) < (j, β, r′).
(i) If Rdri,α is coarser than R
dr
′
j,β then
b(j,β,r′),(i,α,r) = b(i,α,r),(i,α,r).
(ii) If Rdri,α is not coarser than R
dr
′
j,β and l(Rdri,α .R
dr
′
j,β ) > 0 then
b(i,α,r),(j,β,r′) = 0 and b(j,β,r′),(i,α,r) = 0.
(iii) If Rdri,α is coarser than R
dr
′
j,β then
b(i,α,r),(j,β,r′) = 0
where b(i,α,r),(j,β,r′) is the ((i, α, r), (j, β, r′))-th entry after all the
column operations are carried out.
Proof. Part a(i): It follows from equation (2.1), for
b(j,β,r′
1
,r′
2
),(i,α,r1,r2) = a(j,β,r′
1
,r′
2
),(i,α,r1,r2) −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α >d
r′
1
,r′
2
j,β
b(j,β,r′
1
,r′
2
),(l,γ,r′′
1
,r′′
2
)
= a(i,α,r1,r2),(i,α,r1,r2) −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
b(l,γ,r′′
1
,r′′
2
),(l,γ,r′′
1
,r′′
2
)
(by Lemma 2.1 and induction)
= b(i,α,r1,r2),(i,α,r1,r2)
We prove the result by induction on (i, α, r1, r2).
“adm-n2” — 2018/7/24 — 22:32 — page 219 — #57
N. Karimilla Bi, M. Parvathi 219
Let d
0,0
i,α be coarser than d
r′
1
,r′
2
j,β , by lemma 2.1 we have,
l(d0,0i,α.d
0,0
i,α) = l(d0,0i,α.d
r′
1
,r′
2
j,β ) = 0 (2.2)
for any diagram d
r′′
1
,r′′
2
l,γ which is coarser than d
r′
1
,r′
2
j,β and d
r′′
1
,r′′
2
l,γ but not
coarser than d
0,1
i,α, we have
b(i,α,0,1),(l,γ,r′
1
,r′
2
) = 0.
Thus, by applying the column operations L(j,β,r′
1
,r′
2
) → L(j,β,r′
1
,r′
2
) −
L((i,α,0,1) and equation (2.1) ((i, α, 0, 0), (j, β, r′1, r
′
2))-entry becomes
b(i,α,0,0),(j,β,r′
1
,r′
2
) = a(i,α,0,0),(j,β,r′
1
,r′
2
) − a(i,α,0,0),(i,α,0,0) = 1− 1 = 0
by equation (2.2).
(ii) Suppose d
0,1
i,α and d
0,1
j,β such that ♯p(d0,1i,α.d
0,1
j,β) = 2s1 + s2 then by
Lemma 2.1 l(d0,1i,α.d
0,1
r1,r2) = 0 then there exists a unique diagram d
0,0
k,δ
coarser than both d
0,1
i,α and d
0,1
j,β such that
l(d0,0k,δ.d
0,0
k,δ) = l(d0,0k,δ.d
0,1
i,α) = l(d0,0k,δ.d
0,1
j,β) = 0.
Thus, when the column operation L(j,β,0,1) → L(j,β,0,1)−L(k,δ,0,0) is carried
out,
b(i,α,0,1),(j,β,0,1) = a(i,α,0,1),(j,β,0,1) − a(i,α,0,1),(k,δ,0,0) = 1− 1 = 0. (2.3)
Part a(ii): In general, Let d
r1,r2
i,α be not coarser than d
r′
1
,r′
2
j,β such that
l(dr1,r2i,α .d
r′
1
,r′
2
j,β ) > 0. Then by Lemma 2.2 there is a unique diagram d
r′′
1
,r′′
2
k,δ
coarser than both d
r1,r2
i,α and d
r′
1
,r′
2
j,β such that
l(d
r′′
1
,r′′
2
k,δ .d
r′′
1
,r′′
2
k,δ ) = l(d
r′′
1
,r′′
2
k,δ .d
r1,r2
i,α ) = l(d
r′′
1
,r′′
2
k,δ .d
r′
1
,r′
2
j,β )
When the column operations are carried out inductively,
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = a(i,α,r1,r2),(j,β,r′1,r
′
2
) −
∑
d
r′′′
1
,r′′′
2
l,γ
>d
r1,r2
i,α
d
r′′′
1
,r′′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(i,α,r1,r2),(l,γ,r′′′1
,r′′′
2
)
−
∑
d
r′′′′
1
,r′′′′
2
k′,δ′
>d
r′
1
,r′
2
j,β
d
r′′′′
1
,r′′′′
2
k′,δ′
6>d
r1,r2
i,α
b(i,α,r1,r2),(k′,δ′,r′′′′1
,r′′′′
2
)
“adm-n2” — 2018/7/24 — 22:32 — page 220 — #58
220 Gram matrices and Stirling numbers
By induction hypothesis, each entry in the second summation becomes
zero. Thus, we have
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = a(i,α,r1,r2),(j,β,r′1,r
′
2
) −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
).
Also, by induction,
b(i,α,r1,r2),(i′,α′,r′′′′
1
,r′′′′
2
) = b(i′,α′,r′′′′
1
,r′′′′
2
),(i′,α′,r′′′′
1
,r′′′′
2
). (2.4)
Thus,
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = (a(i,α,r1,r2),(j,β,r′1,r′2)
−
∑
d
r′′′
1
,r′′′
2
l,γ
>d
r1,r2
i,α
d
r′′′
1
,r′′′
2
l,γ
>d
r′
1
,r′
2
j,β
d
r′′′
1
,r′′′
2
l,γ
6=d
r′′
1
,r′′
2
k,δ
b(i,α,r1,r2),(l,γ,r′′′1
,r′′′
2
))− b(k,δ,r′′
1
,r′′
2
),(k,δ,r′′
1
,r′′
2
)
= b(k,δ,r′′
1
,r′′
2
),(k,δ,r′′
1
,r′′
2
) − b(k,δ,r′′
1
,r′′
2
),(k,δ,r′′
1
,r′′
2
)
= b(k,δ,r′′
1
,r′′
2
),(k,δ,r′′
1
,r′′
2
) − b(k,δ,r′′
1
,r′′
2
),(k,δ,r′′
1
,r′′
2
)
(by equation (2.4))
= 0
Thus, ((i, α, , r1, r2), (j, β, r
′
1, r
′
2))-entry becomes zero after applying
the column operations when d
r1,r2
i,α is not coarser than d
r′
1
,r′
2
j,β such that
l(dr1,r2i,α .d
r′
1
,r′
2
j,β ) > 0. Also,
b(j,β,r′
1
,r′
2
),(i,α,r1,r2) = a(j,β,r′
1
,r′
2
),(i,α,r1,r2) −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(j,β,r′
1
,r′
2
),(l,γ,r′′
1
,r′′
2
).
since b(j,β,r′
1
,r′
2
),(k,δ,r′′′
1
,r′′′
2
) becomes zero by induction for all d
r′′′
1
,r′′′
2
k,δ coarser
than d
r1,r2
i,α and not coarser than d
r′
1
,r′
2
j,β arguing as in the proof of (ii),
b(j,β,r′
1
,r′
2
),(i,α,r1,r2) = 0.
Part a(iii): In general, let d
r1,r2
i,α be coarser than d
r′
1
,r′
2
j,β , by Lemma 2.1
l(dr1,r2i,α .d
r1,r2
i,α ) = l(dr1,r2i,α .d
r′
1
,r′
2
j,β ) = 2r1 + r2.
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N. Karimilla Bi, M. Parvathi 221
By induction hypothesis,
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = a(i,α,r1,r2),(j,β,r′1,r
′
2
) −
∑
d
r′′
1
,r′′
2
l,γ
>d
i,α
r1,r2
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
)
(2.5)
and
b(i,α,r1,r2),(i,α,r1,r2) = a(i,α,r1,r2),(i,α,r1,r2) −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
)
(2.6)
Thus, when the column operation L(j,β,r′
1
,r′
2
) → L(j,β,r′
1
,r′
2
) − L(i,α,r1,r2)
is carried out the ((i, α, r1, r2), (j, β, r
′
1, r
′
2))-th entry of the block matrix
A2r1+r2,2r′1+r′
2
becomes zero. That is, b(i,α,r1,r2),(j,β,r′1,r′2) = 0.
The proof of (b) is similar to the proof of (a).
Theorem 2.4. (a) After applying the column operations the diagonal entry
x2r1+r2 in the block matrix A2r1+r2,2r1+r2 for 0 6 r1 + r2 6 k − s1 − s2
and the block matrix
−→
A 2r1+r2,2r1+r2 for 0 6 r1 + r2 6 k − s1 − s2 − 1 of
the algebra of Z2-relations and signed partition algebras respectively are
replaced by
(i)
r1−1∏
j=0
[x2 − x− 2(s1 + j)]
r2−1∏
l=0
[x− (s2 + l)] if r1 > 1 and r2 > 1,
(ii)
r2−1∏
j=0
[x− (s2 + j)] if r1 = 0 and r2 6= 0,
(iii)
r1−1∏
j=0
[x2 − x− 2(s1 + j)] if r1 6= 0 and r2 = 0.
Also, the diagonal elements in the block matrix A2r1+r2,2r1+r2 and
−→
A 2r1+r2,2r1+r2 are the same.
(b) After applying the column operations the diagonal entry xr in the
block matrix Ar,r for 0 6 r 6 k is replaced by
r−1∏
j=0
[x− (s+ j)] if r > 1 and 1 if r = 0.
Also, the diagonal elements in the block matrix Ar,r are the same.
Proof. Part (a)(i): The proof is by induction on the number of horizontal
edges.
“adm-n2” — 2018/7/24 — 22:32 — page 222 — #60
222 Gram matrices and Stirling numbers
Let dr1,r2i,α be any diagram corresponding to the diagonal entry x2r1+r2
in block matrix A2r1+r2,2r1+r2 having 2s1 + s2 number of through classes
and r1 pairs of {e}-horizontal edges and r2 number of Z2-horizontal edges.
After applying column operations as mentioned earlier to eliminate
the entries which lie above corresponding to the diagrams coarser than
d
r1,r2
i,α , then by Lemma 2.1 and induction the diagonal entry x2r1+r2 is
replaced as
x2r1+r2 −
∑
06j6r1
−r26j′6r1
−2j+j′<0
B
s1,s2
2r1+r2,2[r1−j]+r2+j′
r1−j−1∏
l=0
[x2 − x− 2(s1 + l)]
×
r2+j′−1∏
f=0
[x− (s2 + f)]
(2.7)
where B
s1,s2
2r1+r2,2p1+p2
gives the number of diagrams which has p1 pairs of
{e} horizontal edges and p2 number of Z2 horizontal edges which lie above
and coarser than d
r1,r2
i,α .
Fix s and put
H2r1+r2,s = −
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′B
s1,s2
2r1+r2,2[r1−j]+r2+j′
C2[r1−j]+r2+j′,s
(2.8)
where C2r′
1
+r′
2
,s denote the coefficient of xs in
r′
1
−1∏
j=0
[x2 − x− 2(s1 + j)]
r′
2
−1∏
l=0
[x− (s2 − l)]
where m = 2r1 + r2 − s.
We shall claim that,
H2r1+r2,s = (−1)mC2r1+r2−1,s.
We shall prove this by using induction on 2r1 + r2.
H2r1+r2,s = −
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′B
s1,s2
2r1+r2,2[r1−j]+r2+j′
C2[r1−j]+r2+j′,s
where m = 2r1 + r2 − s.
“adm-n2” — 2018/7/24 — 22:32 — page 223 — #61
N. Karimilla Bi, M. Parvathi 223
By using Lemma 3.16 in [6] and induction hypothesis, equation (2.8)
becomes,
H2r1+r2,s = −
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′
{
B
s1,s2
2r1+r2−1,2[r1−j]+r2+j′−1
+ (s2 + r2 + j′)Bs1,s2
2r1+r2−1,2[r1−j]+r2+j′
}
×
{
C2[r1−j]+r2+j′−1,s−1 + (s2 + r2 + j′ − 1)C2[r1−j]+r2+j′−1,s
}
The equation (2.8) can be rewritten as follows:
H2r1+r2,s
= −
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′B
s1,s2
2r1+r2−1,2[r1−j]+r2+j′−1C2[r1−j]+r2+j′−1,s−1
−
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′(s2 + r2 + j′ − 1)
×B
s1,s2
2r1+r2−1,2[r1−j]+r2+j′−1C2[r1−j]+r2+j′−1,s
−
∑
06j6r1
−r26j′6r1
−2j+j′<0 and m−2j+j′>0
(−1)2j−j′(s2 + r2 + j′)
×B
s1,s2
2r1+r2−1,2[r1−j]+r2+j′
C2[r1−j]+r2+j′,s
= H2r1+r2−1,s−1 + (−1)m(s2 + r2 − 1)C2r1+r2−1,s
(by canceling common terms)
= (−1)mC2r1+r2−1,s−1 + (−1)m(s2 + r2 − 1)C2r1+r2−1,s (by induction)
Thus, equation (2.8) reduces to
H2r1+r2,s = (−1)mC2r1+r2−1,s−1 + (−1)m(s2 + r2 − 1)C2r1+r2−1,s
= (−1)mC2r1+r2,s
where C2r1+r2,s = C2r1+r2−1,s−1 + (s2 + r2 − 1)C2r1+r2−1,s.
The same proof works for the diagonal element in the block matrix
−→
A 2r1+r2,2r1+r2 for 0 6 r1+r2 6 k−s1−s2−1 in signed partition algebras.
“adm-n2” — 2018/7/24 — 22:32 — page 224 — #62
224 Gram matrices and Stirling numbers
Part (a)(iii): This part can be proved in similar fashion as that of (a)(i)
by using Lemma 3.17 in [6] and
C2r1,s = (−1)mC2(r1−1),s−2 + (−1)mC2(r1−1),s−1
− (−1)m2(s1 + r1 − 1)C2(r1−1),s.
The proof of (b) is same as that of the proof of (a).
Lemma 2.5. Let dr1,r2i,α , d
r′
1
,r′
2
j,β ∈ J2k
2s1+s2
and d
r1,r2
i,α , d
r′
1
,r′
2
j,β ∈
−→
J 2k
2s1+s2
. The
((i, α, r1, r2), (j, β, r
′
1, r
′
2))-entry of the Gram matrices Gk
2s1+s2
of the alge-
bra of Z2-relations and
−→
Gk
2s1+s2
of the signed partition algebras remains
zero even after applying column operations inductively if the Z2-horizontal
edge of dr1,r2i,α coincides with the {e}-through class of d
r′
1
,r′
2
j,β and vice versa.
Proof. The proof follows from Definition 3.7 in [6] and there is no diagram
in common which is coarser than both d
r1,r2
i,α , d
r′
1
,r′
2
j,β ∈ J2r1+r2
2s1+s2
.
Remark 2.6. (a) Let d
r1,r2
i,α , d
r′
1
,r′
2
j,β ∈ J2k
2s1+s2
such that ♯p(dr1,r2i,α .d
r′
1
,r′
2
j,β ) <
2s1 + s2. Place d
r1,r2
i,α above d
r′
1
,r′
2
j,β . Choose sub diagrams dr1−t′
1
,r2−t′
2 ∈
J
2f
2(s1−t1)+s2−t2
of d
r1,r2
i,α and dr
′
1
−t′′
1
,r′
2
−t′′
2 ∈ J
2f
2(s1−t1)+s2−t2
of d
r′
1
,r′
2
j,β such
that
l(dr1−t′
1
,r2−t′
2 .dr
′
1
−t′′
1
,r′
2
−t′′
2 ) > 0
with
♯p((dr1,r2i,α \ dr1−t′
1
,r2−t′
2).(d
r′
1
,r′
2
j,β \ dr
′
1
−t′′
1
,r′
2
−t′′
2 )) < 2t1 + t2.
For the sake of convenience, we shall write
d
r1,r2
i,α = dr1−t′
1
,r2−t′
2 ⊗ d
l1−f
l1−f and d
r′
1
,r′
2
j,β = dr1−t′′
1
,r2−t′′
2 ⊗ d
l2−f
l2−f
where d
l1−f
l1−f = d
r1,r2
i,α \ dr
′
1
−t′′
1
,r′
2
−t′′
2 and d
l2−f
l2−f = d
r′
1
,r′
2
j,β \ dr
′
1
−t′′
1
,r′
2
−t′′
2 .
(b) Let Rdri,α , R
dr
′
j,β ∈ Jk
s such that ♯p(Rdri,α .R
dr
′
j,β ) < 2. Place Rdri,α
above Rdr
′
j,β . Choose sub diagrams Rdr−t′
∈ J
f
s−t of Rdri,α and Rdr
′
−t′′
∈J
f
s−t
of Rdr
′
j,β such that
l(Rdr−t′
.Rdr
′
−t′′
) > 0
with ♯p((Rdri,α \Rdr−t′
).(Rdr
′
j,β \Rdr
′
−t′′
)) < t.
“adm-n2” — 2018/7/24 — 22:32 — page 225 — #63
N. Karimilla Bi, M. Parvathi 225
For the sake of convenience, we shall write
Rdri,α = Rdr−t′
⊗ d
l1−f
l1−f and R
dr
′
j,β = Rdr−t′′
⊗ d
l2−f
l2−f
where d
l1−f
l1−f = Rdri,α \Rdr
′
−t′′
and d
l2−f
l2−f = R
dr
′
j,β \Rdr
′
−t′′
.
Notation 2.7. (a) Let d
r1,r2
i,α , d
r1,r2
j,β be as in Remark 2.6(a) such that
♯p(dr1,r2i,α .d
r1,r2
j,β ) < 2s1 + s2, so that the ((i, α, r1, r2), (j, β, r1, r2))-entry
of the block matrix A2r1+r2,2r1+r2 in algebra of Z2-relations is zero and
0 6 r1 + r2 6 k − s1 − s2.
If t′1 = t′′1 = t1, t
′
2 = t′′2 = t2, put
d
r1,r2
i,α = d
l
f
1
l
f
1
⊗ d
l1−f
l1−f and d
r1,r2
j,β = d
l
f
2
l
f
2
⊗ d
l2−f
l2−f ,
where d
l
f
1
l
f
1
(d
l
f
2
l
f
2
) is the sub diagram of dr1,r2i,α (dr1,r2j,β ), d
l
f
1
l
f
1
, d
l
f
2
l
f
2
∈ J2t1+t2
2t1+t2
and
every {e}-through class (Z2 − through class) of d
l
f
1
l
f
1
is replaced by a {e}-
horizontal edge (Z2 − horizontal edge) and vice versa.
(b) Let dr1,r2i,α , d
r1,r2
j,β be as in Remark 2.6(b) such that ♯p(dr1,r2i,α .d
r1,r2
j,β ) <
2s1 + s2, so that the ((i, α, r1, r2), (j, β, r1, r2))-entry of the block matrix
−→
A 2r1+r2,2r1+r2 in algebra of Z2-relations is zero and 0 6 r1 + r2 6 k −
s1 − s2 − 1.
If t′1 = t′′1 = t1, t
′
2 = t′′2 = t2, put
d
r1,r2
i,α = d
l
f
1
l
f
1
⊗ d
l1−f
l1−f and d
r1,r2
j,β = d
l
f
2
l
f
2
⊗ d
l2−f
l2−f ,
where d
l
f
1
l
f
1
(d
l
f
2
l
f
2
) is the sub diagram of dr1,r2i,α (dr1,r2j,β ), d
l
f
1
l
f
1
, d
l
f
2
l
f
2
∈
−→
J 2t1+t2
2t1+t2
and
every {e}-through class (Z2 − through class) of d
l
f
1
l
f
1
is replaced by a {e}-
horizontal edge (Z2 − horizontal edge) and vice versa.
(c) Let Rdri,α , R
drj,β ∈ Jrs such that ♯p(Rdri,α .R
drj,β ) < s, so that the
((i, α, r), (j, β, r))-entry of the block matrix Ar,r in the partition algebra
is zero and 0 6 r 6 k − s. Put
Rdri,α = dl1l1 ⊗ d
l1−f
l1−f and R
drj,β = dl2l2 ⊗ d
l2−f
l2−f ,
where dl1l1(d
l2
l2
) is the sub diagram of Rdri,α(Rdrj,β ), dl1l1 , d
l2
l2
∈ Jtt and every
through class of dl1l1 is replaced by a horizontal edge and vice versa.
“adm-n2” — 2018/7/24 — 22:32 — page 226 — #64
226 Gram matrices and Stirling numbers
Example 2.8. This example illustrates Notation 2.7.
s.no
1.
2.
d
2,0
i,(2,Φ,2,Φ) d
2,0
j,(2,Φ,2,Φ) d
l
f
1
l
f
1
d
l
f
2
l
f
2
d
l1−f
l1−f = d
l2−f
l2−f
Lemma 2.9. Let (i, α, r1, r2) < (j, β, r′1, r
′
2).
(a) Let d
r1,r2
i,α , d
r′
1
,r′
2
j,β ∈ J2k
2s1+s2
such that ♯p(dr1,r2i,α .d
r′
1
,r′
2
j,β ) < 2s1 + s2
with d
r1,r2
i,α = dr1−t′
1
,r2−t′
2 ⊗ d
l1−f
l1−f and d
r′
1
,r′
2
j,β = dr
′
1
−t′′
1
,r′
2
−t′′
2 ⊗ d
l2−f
l2−f where
dr1−t′
1
,r2−t′
2 , dr
′
1
−t′′
1
,r′
2
−t′′
2 are as in Remark 2.6(a).
(b) Let d
r1,r2
i,α , d
r′
1
,r′
2
j,β ∈
−→
J 2k
2s1+s2
such that ♯p(dr1,r2i,α .d
r′
1
,r′
2
j,β ) < 2s1 + s2
with d
r1,r2
i,α = dr1−t′
1
,r2−t′
2 ⊗ d
l1−f
l1−f and d
r′
1
,r′
2
j,β = dr
′
1
−t′′
1
,r′
2
−t′′
2 ⊗ d
l2−f
l2−f where
dr1−t′
1
,r2−t′
2 , dr
′
1
−t′′
1
,r′
2
−t′′
2 are as in Remark 2.6(a). Then
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = 0,
if any one of the following conditions hold:
(i) 2r1 + r2 < 2r′1 + r′2 or
(ii) if 2r1 + r2 = 2r′1 + r′2 then r1 + r2 < r′1 + r′2 or
(iii) t′′1 6= t1 or t′′2 6= t2 or
(iv) 2r1 + r2 − (2t′1 + t′2) < 2r′1 + r2 − (2t′′1 + t′′2)
(c) Let Rdri,α , R
dr
′
j,β ∈ Jr
′
s such that ♯p(Rdri,α , R
dr
′
j,β ) < s with Rdri,α =
dr−t′ ⊗ Rdri,α \ dr−t′ and R
dr
′
j,β = dr
′−t′′ ⊗ R
dr
′
j,β \ dr
′−t′′ where dr−t′ ∈
Jt
′
t , d
r′−t′′ ∈ Jt
′′
t , Rdri,α \ dr−t′ ∈ Jr−t′
s−t and R
dr
′
j,β \ dr
′−t′′ ∈ Jr
′−t′′
s−t . Then
b(i,α,r),(j,β,r′) = 0,
if any one of the following conditions hold:
(i) r′ < r
(ii) t′′ 6= t
(iii) r − t′ < r′ − t′′
Proof. Part (a): The proof is by induction on the conditions
(i) 2r1 + r2 < 2r′1 + r′2 or
(ii) if 2r1 + r2 = 2r′1 + r′2 then r1 + r2 < r′1 + r′2 or
(iii) t′′1 6= t1 or t′′2 6= t2 or
“adm-n2” — 2018/7/24 — 22:32 — page 227 — #65
N. Karimilla Bi, M. Parvathi 227
(iv) 2r1 + r2 − (2t′1 + t′2) < 2r′1 + r2 − (2t′′1 + t′′2)
Since ♯p(dr1,r2i,α .d
r′
1
,r′
2
j,β ) < 2s1+s2 which implies that a(i,α,r1,r2),(j,β,r′1,r′2) = 0.
After applying column operations inductively we get,
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(l,γ,r′′
1
,r′′
2
),(l,γ,r′′
1
,r′′
2
)
−
∑
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
d
r′′
1
,r′′
2
l,γ
≯d
r1,r2
i,α
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
)
(2.9)
Suppose that ♯p(d
r′′
1
,r′′
2
l,γ .d
r1,r2
i,α ) = 2s1+s2 then by Lemma 2.2 and induction
hypothesis,
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
) = 0.
Suppose that ♯p(d
r′′
1
,r′′
2
l,γ .d
r1,r2
i,α ) < 2s1 + s2 then by using induction on any
one of the conditions (i), (ii), (iii) and (iv)
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
) = 0,
By Lemma 2.2, there exists a unique diagram d
l3−f
l3−f coarser than
both d
l2−f
l2−f and d
l1−f
l1−f and d
l
f
3
l
f
3
∈ J2t1+t2
2t1+t2
which is coarser than dr
′
1
−t′′
1
,r′
2
−t′′
2 .
Denote d
l
f
3
l
f
3
⊗ d
l3−f
l3−f by d
r′′′′
1
,r′′′′
2
k,δ .
It is clear that, d
r′′′′
1
,r′′′′
2
k,δ is coarser than d
r′
1
,r′
2
j,β . Thus, after applying the
column operations L(j,β,r′
1
,r′
2
) → L(j,β,r′
1
,r′
2
) − L(k,δ,r′′′′
1
,r′′′′
2
) we get,
b(i,α,r1,r2),(j,β,r′1,r
′
2
) = −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
d
r′′
1
,r′′
2
l,γ
>d
r′′′′
1
,r′′′′
2
k,δ
b(l,γ,r′′′
1
,r′′′
2
),(l,γ,r′′′
1
,r′′′
2
)
−
∑
d
r′′
1
,r′′
2
l,γ
>d
r′′′′
1
,r′′′′
2
k,δ
d
r′′
1
,r′′
2
l,γ
≯d
r1,r2
i,α
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
) − b(i,α,r1,r2),(k,δ,r′′′′1
,r′′′′
2
)
= b(i,α,r1,r2),(k,δ,r′′′′1
,r′′′′
2
) − b(i,α,r1,r2),(k,δ,r′′′′1
,r′′′′
2
) = 0.
The proof of (b) and (c) are same as that of the proof of (a).
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228 Gram matrices and Stirling numbers
Notation 2.10. Put,
(i) φ
s1,s2
2r1+r2
(x) =
r1−1∏
j=0
[x2−x− 2(s1+ j)]
r2−1∏
l=0
[x− (s2+ l)], r1 > 1, r2 > 1.
(ii) φ
s1,s2
2r1+0(x) =
r1−1∏
j=0
[x2 − x− 2(s1 + j)], r2 = 0.
(iii) φ
s1,s2
2.0+r2
(x) =
r2−1∏
l=0
[x− (s2 + l)], r1 = 0.
(iv) φ
s1,s2
0+0 (x) = 1 and φ
s1,s2
2r1+r2
(x) = 0 if any one of r1, r2 < 0.
(v) φs
r(x) =
r−1∏
l=0
[x− (s+ l)], r > 1
(vi) φs
0(x) = 1 and φs
r = 0 if r < 0.
Now, we derive the following relation between the polynomials which
are needed in the following Lemmas.
Lemma 2.11. We have
(i) φ
s1+t,s2
2(r1−t)+r2
(x) = φ
s1−t,s2
2(r1−t)+r2
(x)
−
2t∑
m=1
2tCmr1−tCm2mm!φs1+t,s2
2(r1−t−m)+r2
(x).
(ii) φ
s1,s2+t
2r1+r2−t(x) = φ
s1,s2−t
2r1+r2−t(x)−
2t∑
m=1
2tCmr2−tCmm!φs1,s2+t
2r1+r2−t−m(x).
(iii) In general,
φ
s1+t1,s2+t2
2(r1−t1)+r2−t2
(x) = φ
s1−t1,s2−t2
2(r1−t1)+r2−t2
(x)
−
2t1∑
k=1
2t1Ck(r1 − t1)Ck2
kk!φs1+t1,s2−t2
2(r1−t1−k)+r2−t2
(x)
−
2t2∑
k′=1
2t2Ck′(r2 − t2)Ck′k
′!φs1−t1,s2+t2
2(r1−t1)+r2−t2−k′
(x)
−
2t1∑
k=1
2t2∑
k′=1
2t1Ck(r1 − t1)Ck2
kk!2t2Ck′(r2 − t2)Ck′k
′!
× φ
s1+t1,s2+t2
2(r1−t1−k)+r2−t2−k′
(x)
where
φ
s1+t,s2
2(r1−t)+r2
(x) =
r1−t−1∏
l=0
[x2 − x− 2(s1 + t+ l)]
r2−1∏
l′=0
[x− (s2 + l′)]
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N. Karimilla Bi, M. Parvathi 229
and
φ
s1,s2+t
2r1+r2−t(x) =
r1−1∏
l=0
[x2 − x− 2(s1 + l)]
r2−t−1∏
l′=0
[x− (s2 + t+ l′)].
Proof. Part (i): We shall prove this by using induction on r1 − t and r2.
Consider
φ
s1−t,s2
2(r1−t)+r2
(x)−
2t∑
m=1
2tCm(r1−t)Cm2mm!φs1+t,s2
2[r1−t−m]+r2
(x) (2.10)
= φ
s1−t,s2
2(r1−t−1)+r2
(x)(x2 − x− 2(s1 + r1 − 2t− 1))
−
2t∑
m=1
2tCm(r1−t)Cm2mm!φs1+t,s2
2(r1−t−m)+r2
(x)
= (φs1−t,s2
2(r1−t−1)+r2
(x) +
2t∑
m=1
2tCm(r1−t−1)Cm2mm!φs1+t,s2
2(r1−t−m−1)+r2
(x))
(x2 − x− 2(s1 + r1 − 2t− 1))−
2t∑
m=1
2tCm(r1−t)Cm2mm!φs1+t,s2
2(r1−t−m)+r2
(x)
(by induction)
= (φs1−t,s2
2(r1−t−1)+r2
(x) +
2t∑
m=1
2tCm(r1−t−1)Cm2mm!φs1+t,s2
2(r1−t−m−1)+r2
(x))
× (x2 − x− 2(s1 + r1 − 2t− 1))
−
2t∑
m=1
2tCm((r1−t−1)Cm + r1−t−1Cm−1)2
mm!φs1+t,s2
2(r1−t−m−1)+r2
(x)
× (x2 − x− 2(s1 + r1 −m− 1))
= φ
s1+t,s2
2(r1−t−1)+r2
(x)(x2 − x− 2(s1 + r1 − 2t− 1))
−
2t∑
m=1
2tCm(r1−t−1)Cm−12
mm!φs1+t,s2
2(r1−t−m)+r2
(x)
+
2t∑
m=1
2tCm(r1−t−1)Cm2mm!(4t− 2m)φs1+t,s2
2(r1−t−m−1)+r2
(x)
= φ
s1+t,s2
2(r1−t−1)+r2
(x)(x2 − x− 2(s1 + r1 − 2t− 1))− 4tφs1+t,s2
2(r1−t−1)+r2
(x)
(by canceling the common terms)
= φ
s1+t,s2
2(r1−t−1)+r2
(x)(x2 − x− 2(s1 + r1 − 1)) = φ
s1+t,s2
2(r1−t)+r2
(x)
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230 Gram matrices and Stirling numbers
Proof of (ii) is similar to the proof of (i) and proof of (iii) follows from
(i) and (ii).
Lemma 2.12. (a) After performing the column operations to eliminate the
non-zero entries corresponding to the diagrams coarser than both d
r1,r2
i,α and
d
r1,r2
j,α , the zero in the ((i, α, r1, r2), (j, β, r1, r2)) entry of the block matrix
A2r1+r2,2r1+r2 for 0 6 r1 + r2 6 k − s1 − s2 in algebra of Z2-relations is
replaced by
−2t1t1!t2!x
2(r1−t1)+r2−t2
where d
r1,r2
i,α and d
r1,r2
j,β are as in Notation 2.7(a).
(b) After performing the column operations to eliminate the non-
zero entries corresponding to the diagrams coarser than both d
r1,r2
i,α and
d
r1,r2
j,β , the zero in the ((i, , α, r1, r2), (j, β, r1, r2)) entry of the block matrix
−→
A 2r1+r2,2r1+r2 for 0 6 r1, r2, r1+r2 6 k−s1−s2−1 in the signed partition
algebra is replaced by
−2t1t1!t2!x
2(r1−t1)+r2−t2 .
where d
r1,r2
i,α and d
r1,r2
j,β are as in Notation 2.7(b).
(c) After performing the column operations to eliminate the non-zero
entries corresponding to the diagrams coarser than both Rdri,α and R
drj,β , the
zero in the ((i, α, r), (j, β, r)) entry of the block matrix Ar,r for 0 6 r 6 k−s
in partition algebra is replaced by
−t!xr−t.
where Rdri,α and R
drj,β are as in Notation 2.7(c).
Proof. Part (a): We shall prove this by induction on t1 and t2.
Case (i): Let t1 = 1 and t2 = 1. We know that the diagrams coarser than
both d
r1,r2
i,α and d
r1,r2
j,β are obtained if and only if the pair of {e}-through
classes and the pair of {e}-horizontal edges of d
l
f
1
l
f
1
or d
l
f
2
l
f
2
is connected by
an {e}-horizontal edge which can be done in two ways and Z2 horizontal
edge and Z2-through class of d
l
f
1
l
f
1
or d
l
f
2
l
f
2
is connected by a Z2-edge which
can be done in one way. Also d
l1−f
l1−f and d
l2−f
l2−f have 2(r1 − 1) + r2 − 1
horizontal edges then after performing the column operations the zero in
the ((i, α, r1, r2), (j, β, r1, r2))-entry of the block matrix A2r1+r2,2r1+r2 is
replaced by
−2
r1−1∑
l=0
r2−1+l∑
l′=0
B
s1,s2
2(r1−1)+r2−1,2(r1−1−l)+r2−1+l′
φ
s1,s2
2[r1−1−l]+r2−1+l′
(x)
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N. Karimilla Bi, M. Parvathi 231
which is equal to
− 2φs1,s2
2[r1−1]+r2−1(x)
− 2
r1−1∑
l=1
r2−1+l∑
l′=1
B
s1,s2
2(r1−1)+r2,2(r1−1−l)+r2−1+l′
φ
s1,s2
2[r1−1−l]+r2−1+l′
(x)
By Theorem 2.4 we know that,
φ
s1,s2
2[r1−1]+r2−1(x) = x2(r1−1)+r2−1
−
r1−1∑
l=1
r2−1+l∑
l′=1
B
s1,s2
2(r1−1)+r2−1,2(r1−1−l)+r2−1+l′
φ
s1,s2
2[r1−1−l]+r2−1+l′
(x).
(2.11)
Substituting equation (2.11) in the above expression and canceling the
common terms we get,
−2x2(r1−1)+r2−1.
In general, the diagrams coarser than both d
r1,r2
i,α and d
r1,r2
j,β are ob-
tained if and only if t1 pairs of {e}-through classes (t2 number of (Z2)-
through classes) and t1 pairs of {e}-horizontal edges (t2 number of (Z2)-
horizontal edges) of d
l
f
1
l
f
1
or d
l
f
2
l
f
2
is connected by an {e}-horizontal edges((Z2)-
horizontal edges) which can be done in 2t1t1!t2! ways. Also d
l1−f
l1−f and
d
l2−f
l2−f have 2(r1 − t1) + r2 − t2 horizontal edges then after performing
the column operations to eliminate the non-zero entries correspond-
ing to the diagrams coarser than both d
r1,r2
i,α and d
r1,r2
j,α the zero in the
((i, α, r1, r2), (j, β, r1, r2))-entry of the block matrix A2r1+r2,2r1+r2 is re-
placed by
− 2t1t1!t2!
r1−t1∑
l=0
r2−t2+l∑
l′=0
B
s1,s2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2+l′
× φ
s1,s2
2[r1−t1−l]+r2−t2+l′
(x)
which is equal to
− 2t1t1!t2!(φ
s1,s2
2[r1−t1]+r2−t2
(x)
−
r1−t1∑
l=1
r2−t2+l∑
l′=1
B
s1,s2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2+l′
φ
s1,s2
2[r1−t1−l]+r2−t2+l′
(x)).
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232 Gram matrices and Stirling numbers
Substituting equation (2.11) in the above expression and canceling the
common terms we get,
−2t1t1!t2!x
2(r1−t1)+r2−t2 .
The proof of (b) and (c) are similar to the proof of (a).
Proposition 2.13. (a) For 0 6 r1 + r2 6 k − s1 − s2, after performing
the column operations to eliminate the non-zero entries which lie above
corresponding to the diagrams coarser than d
r1,r2
j,β , then the ((i, α, r1, r2),
(j, β, r1, r2))-entry of the block matrix A2r1+r2,2r1+r2 and
(b) For 0 6 r1 + r2 6 k − s1 − s2 − 1, after performing the column
operations to eliminate the non-zero entries which lie above corresponding
to the diagrams coarser than d
r1,r2
j,β , then the ((i, α, r1, r2), (j, β, r1, r2))-
entry of the block matrix
−→
A 2r1+r2,2r1+r2 for 0 6 r1 + r2, r1, r2 6 k − s1 −
s2 − 1 are replaced as
(i) (−1)t1+t2(t1)!(t2)!2
t1
r1−1∏
j=t1
[x2−x−2(s1+j)]
r2−1∏
l=t2
[x−(s2+ l)] if r1 > 1
and r2 > 1,
(ii) (−1)t2(t2)!
r2−1∏
l=t2
[x− (s2 + l)] if r1 = 0 and r2 6= 0,
(iii) (−1)t1(t1)!2
t1
r1−1∏
j=t1
[x2 − x− 2(s1 + j)] if r1 6= 0 and r2 = 0,
where d
r1,r2
j,β is as in Notation 2.7.
(c) After performing the column operations to eliminate the non-zero
entries which lie above corresponding to the diagrams coarser than R
dr
′
j,β ,
then the ((i, α, r), (j, β, r))-entry is replaced by
(−1)tt!
r−1∏
j=t
[x− (s+ l)].
where R
drj,β is as in Notation 2.7.
Proof. Part (a): We shall prove this by using induction on t1, t2, the
number of horizontal edges and the index of the diagram (j, β, r1, r2).
“adm-n2” — 2018/7/24 — 22:32 — page 233 — #71
N. Karimilla Bi, M. Parvathi 233
By Lemma 2.9 the ((i, α, r1, r2), (j, β, r1, r2)) entry b(i,α,r1,r2),(j,β,r1,r2)
is given by
b(i,α,r1,r2),(j,β,r1,r2) = −
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
i,α
d
r′′
1
,r′′
2
l,γ
>d
r′
1
,r′
2
j,β
b(l,γ,r′′
1
,r′′
2
),(l,γ,r′′
1
,r′′
2
)
−
∑
d
r′′
1
,r′′
2
l,γ
>d
r1,r2
j,β
d
r′′
1
,r′′
2
l,γ
≯d
r1,r2
i,α
b(i,α,r1,r2),(l,γ,r′′1 ,r
′′
2
).
(2.12)
Case (i): Let t1 = 0, t2 = 1, r1 = 0, r2 = 1 and d
l1−f
l1−f and d
l2−f
l2−f have
2s1+s2−1 through classes and no horizontal edge. After applying column
operations to eliminate the non-zero entries corresponding to the diagrams
coarser than both d
l1−f
l1−f and d
l2−f
l2−f then by Lemma 2.12 and equation
(2.12) the ((i, α, 0, 1), (j, β, 0, 1))-entry b(i,α,0,1),(j,β,0,1) of the block matrix
A2×0+1,2×0+1 is given by
b(i,α,0,1),(j,β,0,1) = (−1)1! .
Since there is no diagram coarser than d
0,1
j,β alone.
Case (ii): Let t1 = 1, t2 = 0, r1 = 1, r2 = 0 and d
l1−f
l1−f and d
l2−f
l2−f have
2(s1 − 1) + s2 through classes and no horizontal edge. After applying
column operations to eliminate the non-zero entries corresponding to the
diagrams coarser than both d
1,0
i,α and d
1,0
j,β then by Lemma 2.12 and equation
(2.12) the ((i, α, 1, 0), (j, β, 1, 0))-entry b(i,α,1,0),(j,β,1,0) of the block matrix
A2×1+0,2×1+0 is given by
b(i,α,1,0),(j,β,1,0) = (−1)21! .
Since there is no diagram coarser than d
1,0
j,β alone.
In general, suppose that the diagrams d
l1−f
l1−f and d
l2−f
l2−f have 2(s1 −
t1) + s2 − t2 through classes and have r1 − t1 pair of {e}-horizontal edges
and r2 − t2 number of Z2-horizontal edges then after performing column
operations to eliminate the coarser elements of dr1,r2i,α and d
r1,r2
j,β having
t′ pair of {e}-through classes ({e}-horizontal edges) with t′ < t, the 0 in
the ((i, α, r1, r2), (j, β, r1, r2))-entry b(i,α,r1,r2),(j,β,r1,r2) of the block matrix
A2r1+r2,2r1+r2 is replaced by −(t1)!(t2)!2
t1x2(r1−t1)+r2−t2 inductively.
For, 0 6 f ′ 6 t1 and 0 6 f ′′ 6 t2, the number of diagrams obtained by
joining f ′ pairs of {e} through classes (f ′′ numbers of Z2 through classes)
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234 Gram matrices and Stirling numbers
with f ′ pairs of {e}-horizontal edges(f ′′ numbers Z2 horizontal edges) in
d
l
f
2
l
f
2
is given by (t1Cf ′)2(t2Cf ′′)22f
′
f ′!f ′′!. The number of diagrams which
are coarser than d
r1,r2
j,β but not coarser than d
r1,r2
i,α having (r1− t1− l)-pairs
of {e}-horizontal edges and r2 − t2 − l′ number of Z2-horizontal edges is
given by
2t1−2f ′∑
m=0
2t2−2f ′′∑
m′=0
(r1 − t1 − l +m)Cm(2t1 − 2f ′)
× Cm2mm!(r2 − t2 − l′ +m′)Cm′(2t2 − 2f ′′)Cm′
× (m′)!(t1Cf ′)2f ′!2f
′
(t2Cf ′′)2f ′′!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′ . (2.13)
Here (2.13) is obtained by choosing m pairs of {e}-horizontal edges (m′
number of Z2-horizontal edges) for every diagram coarser than d
l2−f
l2−f having
r1− t1− (l−m) pairs of {e}-horizontal edges (r2− t2− (l′−m′) number of
Z2-horizontal edges) and choose m pairs of {e}-connected components(m′
number of Z2-connected components) from d
l
f
2
l
f
2
. Connecting the chosen m
pairs of {e}-horizontal edges from d
l2−f
l2−f to the m pairs of {e}-connected
components of d
l
f
2
l
f
2
by {e}-horizontal edge will give 2mm!(m′)! number of
diagrams having r1− t1− l pairs of {e}-horizontal edges. m and m′ cannot
exceed 2t1 − 2f ′ and 2t2 − 2f ′′ respectively, since d
l
f
2
l
f
2
has 2t1 − 2f ′-pairs
of {e}-components and 2t2 − f ′′ number of Z2-components
b(i,α,r1,r2),(j,β,r1,r2) = −2t1t1!t2!x
2(r1−t1)+r2−t2
− (−1)t1+t2(t1)!(t2)!2
t1
{r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=0
2t2∑
m′=0
2t1Cm
× (r1 − t1 − l +m)Cm2mm!2t2Cm′(r2 − t2 − l′ +m′)Cm′(m′)!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
}
−
t1∑
f ′=0
t2∑
f ′′=0
(f ′,f ′′) 6=(0,0)
(f ′,f ′′) 6=(t1,t2)
r1−t1∑
l=0
r2−t2∑
l′=−l
2t1−2f ′∑
m=0
2t2−f ′′∑
m′=0
(t1Cf ′)22f
′
f ′!(t2Cf ′′)2f ′′!
× (−1)t1−f ′
2t1−f ′
(t1 − f ′)!(−1)t2−f ′′
(t2 − f ′′)!2(t1 − f ′)
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N. Karimilla Bi, M. Parvathi 235
× Cm(r1 − t1 − l +m)Cm2mm!2(t2 − f ′′)Cm′(r2 − t2 − l′ +m′)
× Cm′(m′)!Bs1−t1+f ′,s2−t2+f ′′
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′
× φ
s1+t1−f ′,s2+t2−f ′′
2(r1−t1−l)+r2−t2−l′
(x)
= −2t1t1!t2!x
2(r1−t1)+r2−t2 − (−1)t1+t2(t1)!(t2)!2
t1
×
{r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=0
2t2∑
m′=0
2t1Cm(r1 − t1 − l +m)Cm2mm!2t2
× Cm′(r2 − t2 − l′ +m′)Cm′(m′)!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
}
−
t1∑
f ′=0
t2∑
f ′′=0
(f ′,f ′′) 6=(0,0)
(f ′,f ′′) 6=(t1,t2)
(t1Cf ′)22f
′
f ′!(t2Cf ′′)2f ′′!(−1)t1−f ′
2t1−f ′
(t1 − f ′)!
× (−1)t2−f ′′
(t2 − f ′′)!
{r1−t1∑
l=0
r2−t2∑
l′=−l
2t1−2f ′∑
m=0
2t2−f ′′∑
m′=0
2(t1 − f ′)
× Cm(r1 − t1 − l +m)Cm2mm!2(t2 − f ′′)Cm′(r2 − t2 − l′ +m′)
× Cm′(m′)!Bs1−t1+f ′,s2−t2+f ′′
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′
× φ
s1+t1−f ′,s2+t2−f ′′
2(r1−t1−l)+r2−t2−l′
(x)− φ
s1+t1−f ′,s2+t2−f ′′
2(r1−t1)+r2−t2
(x)
}
= −2t1t1!t2!x
2(r1−t1)+r2−t2 − (−1)t1+t2(t1)!(t2)!2
t1
×
{r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=0
2t2∑
m′=0
2t1Cm(r1 − t1 − l +m)Cm2mm!2t2
× Cm′(r2 − t2 − l′ +m′)Cm′(m′)!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
}
−
t1∑
f ′=0
t2∑
f ′′=0
(t1Cf ′)2(−1)t1−f ′
2f
′
f ′!(−1)t2−f ′′
(t2Cf ′′)2f ′′!2t1−f ′
× (t1 − f ′)!(t2 − f ′′)!x2(r1−t1)+r2−t2
+ (−1)t1+t22t1t1!t2!x
2(r1−t1)+r2−t2 + 2t1t1!t2!x
2(r1−t1)+r2−t2
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236 Gram matrices and Stirling numbers
= (−1)t1+t2(t1)!(t2)!2
t1
{
x2(r1−t1)+r2−t2
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=0
2t2∑
m′=0
2t1Cm(r1 − t1 − l +m)Cm2mm!
× 2t2Cm′(r2 − t2 − l′ +m′)Cm′(m′)!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
}
expanding and using Lemma 2.12 we get,
= (−1)t1+t2(t1)!(t2)!2
t1
{
x2(r1−t1)+r2−t2
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
B
s1−t1,s2−t2
2(r1−1)+r2−t2,2(r1−t1−l)+r2−t2−l′
φ
s1−t1,s2−t2
2(r1−t1−l)+r2−t2−l′
(x)
(putting m = 0,m′ = 0)
+
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t2∑
k′=1
2t2Ck′(r2 − t2 − l′)Ck′k
′!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2−l′
φ
s1−t1,s2+t2
2(r1−t1−l)+r2−t2−l′−k′
(x)
(2.14)
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t2∑
m′=1
2t2Cm′(r2 − t2 − l′ +m′)Cm′m′!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2−l′+m′φ
s1−t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
(2.15)
+
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
k=1
2t1Ck(r1 − t1 − l)Ck2
kk!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2−l′
φ
s1+t1,s2−t2
2(r1−t1−l−k)+r2−t2−l′
(x)
(2.16)
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=1
2t1Cm(r1 − t1 − l +m)Cm2mm!
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N. Karimilla Bi, M. Parvathi 237
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′
φ
s1+t1,s2−t2
2(r1−t1−l)+r2−t2−l′
(x)
(2.17)
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
k=1
2t2∑
k′=1
2t1Ck(r1−t1−l)Ck2
kk!2t2Ck′(r2 − t2 − l′)Ck′k
′!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2−l′
φ
s1+t1,s2+t2
2(r1−t1−l−k)+r2−t2−l′−k′
(x)
(2.18)
−
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=1
2t2∑
k′=1
2t1Cm(r1 − t1 − l +m)Cm2mm!2t2
× Ck′(r2 − t2 − l′)Ck′k
′!Bs1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′
× φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′−k′
(x) (2.19)
+
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
k=1
2t2∑
m′=1
2t1Ck(r1−t1−l)Ck2
kk!2t2Cm′(r2−t2−l′+m′)
× Cm′m′!Bs1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l)+r2−t2−l′+m′
× φ
s1+t1,s2+t2
2(r1−t1−l−k)+r2−t2−l′
(x) (2.20)
+
r1−t1∑
l=0
r2−t2∑
l′=−l
(l,l′) 6=(0,0)
2t1∑
m=1
2t2∑
m′=1
2t1Cm(r1 − t1 − l +m)Cm2mm!2t2
× Cm′(r2 − t2 − l′ +m′)Cm′m′!
×B
s1−t1,s2−t2
2(r1−t1)+r2−t2,2(r1−t1−l+m)+r2−t2−l′+m′φ
s1+t1,s2+t2
2(r1−t1−l)+r2−t2−l′
(x)
(2.21)
Putting (l = 0, l′ = m′) in equation (2.15), (l = m, l′ = 0) in equation
(2.17), (l = m, l′ = m′) in equation (2.21) and canceling the common
terms , we get
b(i,α,r1,r2),(j,β,r1,r2)
= (−1)t1+t2(t1)!(t2)!2
t1
{
φ
s1−t1,s2−t2
2(r1−t1)+r2−t2
(x)
−
2t1∑
m=1
2t1Cm(r1 − t1)Cm2mm!φs1+t1,s2−t2
2(r1−t1−m)+r2−t2
(x)
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238 Gram matrices and Stirling numbers
−
2t2∑
m′=1
2t2Cm′(r2 − t2)Cm′m′!φs1−t1,s2+t2
2(r1−t1)+r2−t2−m′(x)
−
2t1∑
m=1
2t2∑
m′=1
2t1Cm(r1 − t1)Cm2mm!2t2Cm′(r2 − t2)Cm′m′!
× φ
s1+t1,s2+t2
2(r1−t1−m)+r2−t2−m′(x)
}
= (−1)t1+t2(t1)!(t2)!2
t1φ
s1+t1,s2+t2
2(r1−t1)+r2−t2
(x)
Therefore the ((i, α, r1, r2), (j, β, r
′
1, r
′
2))-entry in the block matrix
A2r1+r2,2r1+r2 is replaced as
(−1)t1+t2(t1)!(t2)!2
t1
r1−1∏
l=t1
[x2 − x− 2(s1 + l)]
r2−1∏
m=t2
[x− (s2 +m)].
Proof (b) and (c) are similar to the proof of (a).
Now, we have the main theorem of this section.
2.2. Main theorem
Theorem 2.14. (a) Let G̃k
2s1+s2
be the matrix similar to the Gram matrix
Gk
2s1+s2
of the algebra of Z2-relations which is obtained after the column
operations and the corresponding row operations on Gk
2s1+s2
. Then
G̃k
2s1+s2
=
( ⊕
06r1+r26k−s1−s2
Ã2r1+r2,2r1+r2
)
(b) Let
−̃→
G
k
2s1+s2
be the matrix similar to the Gram matrix
−→
Gk
2s1+s2
of
signed partition algebras which is obtained after the column operations and
the corresponding row operations on
−→
Gk
2s1+s2
. Then
−̃→
G
k
2s1+s2
=
( ⊕
06r16k−s1−s2−1
06r2<k−s1−s2−1
2r1+r262k−2s1−2s2−1
−̃→
A 2r1+r2,2r1+r2
)⊕ −̃→
A ρ
where
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N. Karimilla Bi, M. Parvathi 239
(i) the diagonal element of Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2 are given
by
1.
r1−1∏
j=0
[x2 − x− 2(s1 + j)]
r2−1∏
l=0
[x− (s2 + l)] if r1 > 1, r2 > 1;
2.
r1−1∏
j=0
[x2 − x− 2(s1 + j)] if r2 = 0;
3.
r2−1∏
l=0
[x− (s2 + l)] if r1 = 0.
(ii) The entry b(i,α,r1,r2),(j,β,r1,r2) of the block matrix Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2 are replaced by
1. (−1)t1+t22t1(t1)!(t2)!
r1−t1−1∏
j=0
[x2 − x− 2(s1 + t1 + j)]
×
r2−t2−1∏
l=0
[x− (s2 + t2 + l)] if r1 > 1, r2 > 1;
2. (−1)t12t1(t1)!
r1−t1−1∏
j=0
[x2 − x− 2(s1 + t1 + j)] if r2 = 0;
3. (−1)t2(t2)!
r2−t2−1∏
l=0
[x− (s2 + t2 + l)] if r1 = 0.
whenever d
r1,r2
i,α and d
r1,r2
j,β is as in Remark 2.6(a), Notation 2.7 and
Proposition 2.13.
(iii) All other entries in the block matrix Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2
are zero.
The underlying partitions of the diagrams corresponding to the en-
tries of the block matrix
−̃→
A 2r1+r2,2r1+r2 are α = [α1]
1[α2]
2[α3]
3[α4]
4 with
α1 = (α11, . . . , α1s1), α2 = (α21, . . . , α2s2), α3 = (α31, . . . , α3r1), α4 =
(α41, . . . , α4r2) such that atleast one of α1i, α2j , α3l, α4m is greater than 1
for 1 6 i 6 s1, 1 6 j 6 s2, 1 6 l 6 r1 and 1 6 m 6 r2.
Since the diagrams corresponding to the partition ρ=[ρ1]
1[ρ2]
2[ρ3]
3[ρ4]
4
with |ρ1i| = 1 ∀1 6 i 6 s1, |ρ2j | = 1 ∀1 6 j 6 s2, |ρ3m| = 0 ∀1 6 m 6 r1
and |ρ4l| = 1 ∀1 6 l 6 r2 does not belong to the signed partition algebra.
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240 Gram matrices and Stirling numbers
Thus the block corresponding to the diagrams whose underlying partition
is ρ is studied separately.
(b)′ Let
−̃→
A ρ where the partition ρ is such that each ρ1i, ρ2j , ρ3l, ρ4m is
equal to 1 for 1 6 i 6 s1, 1 6 j 6 s2, 1 6 l 6 r1 and 1 6 m 6 r2 and
−̃→
A ρ
is the block sub matrix corresponding to the diagrams whose underlying
partition is ρ.
(i) The ((i, ρ, r′1, r
′
2), (i, ρ, r
′
1, r
′
2))-entry x2r
′
1
+r′
2 in the matrix
−̃→
A ρ is
replaced by
r′
1
−1∏
j=0
[x2 − x− 2(s1 + j)]
l=r′
2
−1∏
l=0
[x− (s2 + l)] +
k−s1−s2−1∏
l=0
[x− (s2 + l)]
where 1 6 r′1 6 k − s1 − s2 and r′2 = k − s1 − s2 − r′1.
(ii) The zero in the ((i, ρ, r′1, r
′
2), (j, ρ, r
′
1, r
′
2))-entry in the block matrix
−̃→
A ρ is replaced by
(−1)t1+t22t1(t1)!(t2)!
r′
1
−t1−1∏
j=0
[x2 − x− 2(s1 + t1 + j)]
×
r′
2
−t2−1∏
l=0
[x− (s2 + l + t2)] +
k−s1−s2−1∏
l=0
[x− (s2 + l)]
where d
r′
1
,r′
2
i,ρ and d
r′
1
,r′
2
j,ρ are as in Remark 2.6(a), Notation 2.7 and
Proposition 2.13 where 1 6 i, j 6 2k − 2s1 − 2s2 and i 6= j.
(iii) If ♯p(d
r′
1
,k−s1−s2−r′
1
i,ρ .d
r1,k−s1−s2−r1
j,ρ ) = 2s1 + s2 then the ((i, ρ, r′1, k−
s1 − s2 − r′1), (j, ρ, r1, k − s1 − s2 − r1))-entry in the block matrix
−̃→
A ρ is replaced by
(−1)r1+r′
1
k−s1−s2−1∏
l=0
[x− (s2 + l)]
where 1 6 i, j 6 2k − 2s1 − 2s2 and i 6= j.
(iv) All other entries in the block matrix
−̃→
A ρ are zero.
(c) Let G̃k
s be the matrix similar to the Gram matrix Gk
s which is
obtained after the column operations and the row operations on Gk
s . Then
G̃k
s =
( ⊕
06r6k−s
Ãr,r
)
“adm-n2” — 2018/7/24 — 22:32 — page 241 — #79
N. Karimilla Bi, M. Parvathi 241
where
(i) The diagonal element of Ãr,r is given by
r−1∏
l=0
[x− (s+ l)]
(ii) The entry b(i,α,r),(j,β,r) of the block matrix Ãr,r is replaced by
(−1)t(t)!
r−1∏
j=t
[x− (s+ l)]
whenever Rdri,α and R
drj,β are as in Remark 2.6(b), Notation 2.7 and
Proposition 2.13.
(iii) All other entries in the block matrix
−→
A r,r are zero.
Proof. Part (a): Every entry x2r1+r2 in the sub block matrix Ã2r1+r2,2r′1+r′
2
is also replaced by
r1−1∏
j=0
[x2 − x− 2(s1 + j)]
r2−1∏
l=0
[x− (s2 + l)]
We continue to do the column operations for all the diagrams
whose underlying partition is α where α = [α1]
1[α2]
2[α3][α4]
4 with
α1 = (α11, . . . , α1s1), α2 = (α21, . . . , α2s2), α3 = (α31, . . . , α3r1), α4 =
(α41, . . . , α4r2) such that at least one of α1i, α2j , α3l, α4m is greater than 1
and hence the above entry gets eliminated.
Thus, from Lemmas 2.1 and 2.9 it follows that the rectangular sub
matrix Ã2r1+r2,2r′1+r′
2
with 2r1 + r2 6= 2r′1 + r′2 becomes zero after all the
column operations are carried out.
After applying the row operations corresponding to the column opera-
tions performed in Lemmas 2.5, 2.9, Proposition 2.13, and Theorem 2.4,
the Gram matrix Gk
2s1+s2
which is similar to a matrix G̃k
2s1+s2
decomposes
as a direct sum of block matrices, i.e.
G̃k
2s1+s2
=
( ⊕
06r1+r26k−s1−s2
Ã2r1+r2,2r1+r2
)
where the diagonal element of Ã2r1+r2,2r1+r2 is given by
r1−1∏
j=0
[x2 − x− 2(s1 + j)]
r2−1∏
l=0
[x− (s2 + l)].
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242 Gram matrices and Stirling numbers
Result (i) follows from Theorem 2.4(a), result (ii) follows from Proposition
2.13(a) and result (iii) follow from Lemmas 2.3, 2.5, and 2.9(a) respectively.
Part (b)′: The column operations corresponding to the diagrams whose
underlying partition is ρ where
ρ = [ρ1]
1[ρ2]
2[ρ3]
3[ρ4]
4 where |ρ1i| = 1, ∀1 6 i 6 s1, |ρ2j | = 1∀1 6
j 6 s2, |ρ3m| = 0, ∀1 6 m 6 r1 and |ρ4l| = 1∀1 6 l 6 r2 such that
s1 + s2 + r2 = k with s1 � k cannot be carried out for the block matrix
−̃→
A ρ, since those diagrams do not belong to the signed partition algebra.
Part (i): We prove the result by induction.
Case (i): Let d
1,k−s1−s2−1
i,ρ be a diagram in
−→
J 2.1+k−s1−s2−1
2s1+s2
, after the
column operations the ((i, ρ, 1, k−s1−s2−1), (i, ρ, 1, k−s1−s2−1))-entry
corresponding to the diagram d
1,k−s1−s2−1
i,ρ will be replaced by
φ
s1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x)
since the signed partition algebra does not contain diagrams with k−s1−s2
number of Z2-horizontal edges.
Case (ii): Let d
2,k−s1−s2−2
i,ρ be a diagram in
−→
J 2.2+k−s1−s2−2
2s1+s2
.
After applying the column operations
L(i,ρ,2,k−s1−s2−2) → L(i,ρ,2,k−s1−s2−2) − L(k,α,r1,r2)
for all dr1,r2k,α where d
r1,r2
k,α ∈
−→
J 2r1+r2
2s1+s2
with r1 + r2 + s1 + s2 6 k − 1, the
((i, ρ, 2, k− s1 − s2 − 2), (i, ρ, 2, k− s1 − s2 − 2))-entry will be replaced by
φ
s1,s2
2.2+k−s1−s2−2(x) + 2φs1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x)
Again applying the column operations inside the block matrix
−̃→
A ρ,
the ((i, ρ, 2, k − s1 − s2 − 2), (i, ρ, 2, k − s1 − s2 − 2))-entry becomes
φ
s1,s2
2.2+k−s1−s2−2(x) + 2φs1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x)
− 2
[
φ
s1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x)
]
= φ
s1,s2
2.2+k−s1−s2−2(x)− φ
s1,s2
2.0+k−s1−s2
(x).
After applying the row operations corresponding to the column oper-
ations which is performed to obtain the above ((i, ρ, 2, k − s1 − s2 − 2),
(i, ρ, 2, k− s1− s2− 2))-entry, the ((i, ρ, 2, k− s1− s2− 2), (i, ρ, 2, k− s1−
s2 − 2))-entry further becomes
φ
s1,s2
2.2+k−s1−s2−2(x)− φ
s1,s2
2.0+k−s1−s2
(x) + 2φs1,s2
2.0+k−s1−s2
(x)
= φ
s1,s2
2.2+k−s1−s2−2(x) + φ
s1,s2
2.0+k−s1−s2
(x).
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N. Karimilla Bi, M. Parvathi 243
In general, let d
j,k−s1−s2−j
i,ρ be a diagram in
−→
J 2.j+k−s1−s2−j
2s1+s2
. After
applying the column operations, by induction the ((i, ρ, j, k − s1 − s2 − j),
(i, ρ, j, k − s1 − s2 − j))-entry of the matrix
−̃→
A ρ becomes
φ
s1,s2
2j+k−s1−s2−j(x) +
j−1∑
m=1
jCmφ
s1,s2
2(j−m)+k−s1−s2−j+m
(x) + φ
s1,s2
2.0+k−s1−s2
(x)
−
j−1∑
m=1
jCm(φs1,s2
2(j−m)+k−s1−s2−j+m
(x) + φ
s1,s2
2.0+k−s1−s2
(x))
= φ
s1,s2
2j+k−s1−s2−j(x)−
j−1∑
m=1
jCmφ
s1,s2
2.0+k−s1−s2
(x) + φ
s1,s2
2.0+k−s1−s2
(x)
After applying the row operations the ((i, ρ, j, k − s1 − s2 − j), (i, ρ, j,
k − s1 − s2 − j))-entry further becomes
φ
s1,s2
2j+k−s1−s2−j(x)−
j−1∑
m=1
jCmφ
s1,s2
2.0+k−s1−s2
(x) + φ
s1,s2
2.0+k−s1−s2
(x)
+
j−1∑
m=1
jCmφ
s1,s2
2.0+k−s1−s2
(x)
= φ
s1,s2
2j+k−s1−s2−j(x) + φ
s1,s2
2.0+k−s1−s2
(x)
Thus, for a diagram d
r′
1
,k−s1−s2−r′
1
i,ρ ∈
−→
J
2r′
1
+k−s1−s2−r′
1
2s1+s2
the ((i, ρ, r′1, k−
s1 − s2 − r′1), (i, ρ, r
′
1, k − s1 − s2 − r′1))-entry is replaced as
r′
1
−1∏
j=0
[x2 − x− 2(s1 + j)]
k−s1−s2−r′
1
−1∏
l=0
[x− (s2 + l)] +
k−s1−s2−1∏
l=0
[x− (s2 + l)].
Part (ii): The proof follows from Proposition 2.13(b) and it is similar to
the Proof of (1), whenever d
r′
1
,r′
2
i,ρ and d
r′
1
,r′
2
j,ρ are as in Notation 2.7.
Part (iii). Case (i): Let d1,k−s1−s2−1
i,ρ ∈
−→
J 2.1+k−s1−s2−1
2s1+s2
and d
2,k−s1−s2−2
j,ρ ∈
−→
J 2.2+k−s1−s2−2
2s1+s2
such that number of {e}-horizontal edges in d
1,k−s1−s2−1
i,ρ
is greater than the number of {e}-horizontal edges in d
2,k−s1−s2−2
j,ρ then
l(d1,k−s1−s2−1
i,ρ .d
2,k−s1−s2−2
j,ρ ) 6 2.1 + k − s1 − s2 − 1.
There will be two diagrams say d
1,k−s1−s2−1
i′,ρ and d
1,k−s1−s2−1
i′′,ρ coarser
than d
2,k−s1−s2−2
j,ρ .
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244 Gram matrices and Stirling numbers
Subcase (i): Suppose l(d1,k−s1−s2−1
i,ρ .d
2,k−s1−s2−2
j,ρ ) = 2.1+k−s1−s2−1
then
a(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2) = φ
s1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x).
Also,
a(i,ρ,1,k−s1−s2−1),(i′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x)
and
a(i,ρ,1,k−s1−s2−1),(i′′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.0+k−s1−s2
(x),
or
a(i,ρ,1,k−s1−s2−1),(i′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.0+k−s1−s2
(x)
and
a(i,ρ,1,k−s1−s2−1),(i′′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.1+k−s1−s2−1(x) + φ
s1,s2
2.0+k−s1−s2
(x).
After applying the column operations the ((i, ρ, 1, k − s1 − s2 − 1),
(j, ρ, 2, k − s1 − s2 − 2))-entry in
−̃→
A ρ becomes
b(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2) = a(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2)
− a(i,ρ,1,k−s1−s2−1),(i′,ρ,1,k−s1−s2−1)
− a(i,ρ,1,k−s1−s2−1),(i′′,ρ,1,k−s1−s2−1)
= −φ
s1,s2
2.0+k−s1−s2
(x).
Subcase (ii): Suppose l(d1,k−s1−s2−1
i,ρ .d
2,k−s1−s2−2
j,ρ ) < 2.1+k−s1−s2−1
then
a(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2) = φ
s1,s2
2.0+k−s1−s2
(x).
Also,
a(i,ρ,1,k−s1−s2−1),(i′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.0+k−s1−s2
(x)
and
a(i,ρ,1,k−s1−s2−1),(i′′,ρ,1,k−s1−s2−1) = φ
s1,s2
2.0+k−s1−s2
(x)
After applying the column operations the (i, ρ, 1, k−s1−s2−1), (j, ρ, 2, k−
s1 − s2 − 2)-entry in
−̃→
A ρ becomes
b(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2) = a(i,ρ,1,k−s1−s2−1),(j,ρ,2,k−s1−s2−2)
− a(i,ρ,1,k−s1−s2−1),(i′,ρ,1,k−s1−s2−1)
− a(i,ρ,1,k−s1−s2−1),(i′′,ρ,1,k−s1−s2−1)
= −φ
s1,s2
2.0+k−s1−s2
(x)
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N. Karimilla Bi, M. Parvathi 245
In general, let d
r′
1
,k−s1−s2−r′
1
i,ρ ∈
−→
J
2r′
1
+k−s1−s2−r′
1
2s1+s2
and d
r1,k−s1−s2−r1
j,ρ ∈
−→
J 2r1+k−s1−s2−r1
2s1+s2
such that the number of {e}-horizontal edges in
d
r1,k−s1−s2−r1
j,ρ is strictly greater than the number of {e}-horizontal edges in
d
r′
1
,k−s1−s2−r′
1
i,ρ then l(d
r′
1
,k−s1−s2−r′
1
i,ρ .d
r1,k−s1−s2−r1
j,ρ ) 6 2r′1+k−s1−s2−r′1.
After applying the column operations the ((i, ρ, r′1, k − s1 − s2 − r′1),
(j, ρ, r1, k − s1 − s2 − r1))-entry becomes
b(i,ρ,r′
1
,k−s1−s2−r′
1
),(j,ρ,r1,k−s1−s2−r1)
×
(r1−1∑
m=1
(−1)m−1
r1Cm − 1
)
φ
s1,s2
2.0+k−s1−s2
(x)
= (−1)r1+1φ
s1,s2
2.0+k−s1−s2
(x)
After applying row operations the ((i, ρ, r′1, k − s1 − s2 − r′1), (j, ρ, r1,
k − s1 − s2 − r1))-entry further becomes
b(i,ρ,r′
1
,k−s1−s2−r′
1
),(j,ρ,r1,k−s1−s2−r1)
= (
r′
1
−1∑
m=1
(−1)m−1
r′
1
Cm − 1)(−1)r1+1φ
s1,s2
2.0+k−s1−s2
(x)
= (−1)r
′
1
+r1φ
s1,s2
2.0+k−s1−s2
(x)
Thus, the ((i, ρ, r′1, k − s1 − s2 − r′1), (j, ρ, r1, k − s1 − s2 − r1))-entry
of the block matrix
−̃→
A ρ is given by
(−1)r1+r′
1φ
s1,s2
2.0+k−s1−s2
(x).
The proof of (b) and (c) is similar to the proof of (a).
Remark 2.15. (a) G̃k
0+0 = ⊕
06r1+r26k
Ã2r1+r2,2r1+r2 .
(b)
−̃→
G
k
0+0 = ⊕
06r16k−1
06r26k−1
2r1+r262k−1
−̃→
A
k
2r1+r2,2r1+r2
⊕
−̃→
A ρ, where Ã2r1+r2,2r1+r2 and
−̃→
A 2r1+r2,2r1+r2 are the diagonal block matrices whose diagonal entry is
given by
(i)
r1−1∏
j=0
[x2 − x− 2j]
r2−1∏
l=0
[x− l], r1 > 1, r2 > 1,
(ii)
r2−1∏
l=0
[x− l], r1 = 0,
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246 Gram matrices and Stirling numbers
(iii)
r1−1∏
j=0
[x2 − x− 2j], r2 = 0.
(b)′ Let
−̃→
A ρ where the partition ρ is such that ρ1i = Φ, ρ2j = Φ,
ρ3l = 1, ρ4m = 1 for 1 6 i 6 s1, 1 6 j 6 s2, 1 6 l 6 r1 and 1 6 m 6 r2
and
−̃→
A ρ is the block sub matrix corresponding to the diagrams whose
underlying partition is ρ.
The ((i, ρ, r′1, r
′
2), (i, ρ, r
′
1, r
′
2))-entry x2r
′
1
+r′
2 of the matrix
−̃→
A ρ is re-
placed by
r′
1
−1∏
j=0
[x2 − x− 2j]
r′
2
−1∏
l=0
[x− l] +
r′
1
+r′
2
−1∏
l=0
[x− l].
(c) Gk
0 = ⊕
06r6k
Ãr,r
r−1∏
l=0
[x− l].
3. Semisimplicity of signed partition algebras
Semisimplicity of the algebra of Z2-relations and partition algebras
are already discussed in [15] and [2] respectively. In this paper, we give
an alternate approach to show that the partition algebras and the algebra
of Z2-relations are semisimple. We also study about the semisimplicity of
signed partition algebras.
Definition 3.1. [5] Let s = 2s1 + s2. For 0 6 s 6 2k and ((s, (s1, s2)),
((λ1, λ2), µ)) ∈ Λ′ (((s, (s1, s2)), ((λ1, λ2), µ)) ∈
−→
Λ ′), put λ = (λ1, λ2).
The left cell module
W [(s, (s1, s2)), ((λ1, λ2), µ)] (
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)])
for the cellular algebra A
[
AZ2
k
]
(A
[−→
AZ2
k
]
) is defined as follows:
(i) W [(s, (s1, s2)), ((λ1, λ2), µ)] is a free A-module with basis
{
C
mλ
sλ
m
µ
sµ
S
∣∣∣S = (d, P ) ∈ Mk [(s, (s1, s2))]
}
“adm-n2” — 2018/7/24 — 22:32 — page 247 — #85
N. Karimilla Bi, M. Parvathi 247
and AZ2
k -action is defined on the basis element by a
aC
mλ
sλ
m
µ
sµ
S ≡
∑
(S′,s′)∈M ′k
[(
s,(s1,s2),((λ1,λ2),µ)
)]
C
ra(S′,S)mλ
s′
λ
m
µ
s′µ
S′
mod AZ2
k
(
<
(
s, (s1, s2), ((λ1, λ2), µ)
))
,
where (S,w) = ((d, P ), ((sλ1
, sλ2
), sµ)), (S
′, s′) = ((d′, P ′), ((s′λ1
, s′λ2
), s′µ)),
ra(S
′, S) is as in 3(a)(i) and (b)(i) of Theorem 5.4.
(ii)
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)] is a free A-module with basis
{
−→
C
mλ
sλ
m
µ
sµ
−→
S
∣∣∣−→S = (d, P ) ∈
−→
Mk [(s, (s1, s2))]
}
and
−→
AZ2
k -action is defined on the basis element by −→a
−→a
−→
C
mλ
sλ
m
µ
sµ
S ≡
∑
(S′,s′)∈
−→
M ′k
[(
s,(s1,s2),((λ1,λ2),µ)
)]
−→
C
r−→a (S′,S)mλ
s′
λ
m
µ
s′µ
S′
mod
−→
AZ2
k
(
<
(
s, (s1, s2), ((λ1, λ2), µ)
))
,
where (S,w) = ((d, P ), ((sλ1
, sλ2
), sµ)), (S
′, s′) = ((d′, P ′), ((s′λ1
, s′λ2
), s′µ)),
ra(S
′, S) is as in 3(a)(ii) and (b)(ii) of Theorem 5.4.
Lemma 3.2 ([5]).
(i) C
mλ
sλ,sλ
m
µ
sµ,sµ
S,S C
mλ
tλ,tλ
m
µ
tµ,tµ
T,T ≡ Φ1((S, s), (T, t))C
mλ
sλ,tλ
m
µ
sµ,tµ
S,T
mod
[
AZ2
k < (s, (s1, s2), ((λ1, λ2), µ)
]
where
Φ1((S, s), (T, t))
=
xl(P∨P ′)φλ
δ1
(sλ, tλ)φ
µ
δ2
(sµ, tµ) when conditions (a) and (b)
of Definition 4.6 in [5] hold,
0 otherwise.
(ii)
−→
C
mλ
sλ,sλ
m
µ
sµ,sµ
S,S
−→
C
mλ
tλ,tλ
m
µ
tµ,tµ
T,T ≡
−→
Φ 1((S, s), (T, t))C
mλ
sλ,tλ
m
µ
sµ,tµ
S,T
mod
[−→
AZ2
k < (s, (s1, s2), ((λ1, λ2), µ)
]
“adm-n2” — 2018/7/24 — 22:32 — page 248 — #86
248 Gram matrices and Stirling numbers
where
−→
Φ 1((S, s), (T, t))
=
xl(P∨P ′)φλ
δ1
(sλ, tλ)φ
µ
δ2
(sµ, tµ) when conditions (a) and (b)
of Definition 4.6 in [5] hold,
0 otherwise.
Here (S, s) = ((d, P ), ((sλ1
, sλ2
), sµ)), (T, t) = ((d′, P ′), ((tλ1
, tλ2
), tµ)),
and l(P ∨ P ′)(l(P ∨ P ′)) denotes the number of connected components in
d′.d′′ excluding the union of all the connected components of P and P ′,
mλ
sλ,sλ
δ1m
λ
tλ,tλ
≡ φλ
δ1
(sλ, tλ)m
λ
s′
λ
,tλ
mod H(< (λ1, λ2)),m
µ
sµ,sµδ2m
µ
tµ,tµ
≡ φ
µ
δ2
(sµ, tµ)m
µ
s′µ,tµ
mod H
′(< µ)
as in Lemma 1.7 in [1].
Definition 3.3 ([5]). For
(s, (s1, s2), ((λ1, λ2), µ)) ∈ Λ′ (s, (s1, s2), ((λ1, λ2), µ)) ∈
−→
Λ ′
the bilinear map φ
λ,µ
s1,s2(
−→
φ
λ,µ
s1,s2) is defined as
(i) φλ,µ
s1,s2
(
C
mλ
sλ,sλ
m
µ
sµ,sµ
(d,P ) , C
mλ
tλ,tλ
m
µ
tµ,tµ
(d′,P ′)
)
= Φ1((S, s), (T, t)),
(S, s), (T, t) ∈ M ′k
[
s, (s1, s2), ((λ1, λ2), µ)
]
;
(ii)
−→
φ λ,µ
s1,s2
(−→
C
mλ
sλ,sλ
m
µ
sµ,sµ
(d,P ) ,
−→
C
mλ
tλ,tλ
m
µ
tµ,tµ
(d′,P ′)
)
= Φ1((S, s), (T, t)),
(S, s), (T, t) ∈
−→
M ′k
[
s, (s1, s2), ((λ1, λ2), µ)
]
,
where Φ1((S, s), (T, t))(
−→
Φ 1((S, s), (T, t))) is as in Lemma 3.2.
Put
(i) G
λ,µ
2s1+s2
= (Φ1((S, s), (T, t)))(S,s),(T,t)∈M ′k
[
s,(s1,s2),((λ1,λ2),µ)
],
where
Φ1((S, s), (T, t))
=
xl(Pi∨Pj)φλ
δ1
(sλ, tλ)φ
µ
δ2
(sµ, tµ) when conditions (a) and (b)
of Definition 4.6 in [5] hold,
0 otherwise,
where (S, s) = ((di, Pi), ((sλ1
, sλ2
), sµ)), (T, t) = ((dj , Pj), ((tλ1
, tλ2
), tµ)).
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N. Karimilla Bi, M. Parvathi 249
(ii)
−→
G
λ,µ
2s1+s2
= (
−→
Φ 1((S, s), (T, t)))(S,s),(T,t)∈−→M ′k
[
s,(s1,s2),((λ1,λ2),µ)
],
where
−→
Φ 1((S, s), (T, t))
=
xl(Pi∨Pj)φλ
δ1
(sλ, tλ)φ
µ
δ2
(sµ, tµ) when conditions (a) and (b)
of Definition 4.6 in [5] hold,
0 otherwise,
where (S, s) = ((di, Pi), ((sλ1
, sλ2
), sµ)), (T, t) = ((dj , Pj), ((tλ1
, tλ2
), tµ)),
and l(Pi ∨ Pj) denotes the number of connected components in d′.d′′
excluding the union of all the connected components of Pi and Pj ,
mλ
sλ,sλ
δ1m
λ
tλ,tλ
≡ φλ
δ1
(sλ, tλ)m
λ
s′
λ
,tλ
mod H(< (λ1, λ2))
and
mµ
sµ,sµδ2m
µ
tµ,tµ
≡ φ
µ
δ2
(sµ, tµ)m
µ
s′µ,tµ
mod H
′(< µ)
as in Lemma 1.7 in [1].
G
λ,µ
2s1+s2
(
−→
G
λ,µ
2s1+s2
) is called the Gram matrix of the cell module
W [(s, (s1, s2)), ((λ1, λ2), µ)] (
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)).
Definition 3.4. Let
{
C
mλ
sλ
m
µ
sµ
S
r1,r2
i,α
}
(S
r1,r2
i,α ,tl)∈M ′k[(s,(s1,s2)),((λ1,λ2),µ)]
({−→
C
mλ
sλ
m
µ
sµ
S
r1,r2
i,α
}
(S
r1,r2
i,α ,tl)∈
−→
M ′k[(s,(s1,s2)),((λ1,λ2),µ)]
)
be the basis of the cell module
W [(s, (s1, s2)), ((λ1, λ2), µ)] (
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)]),
where S
r1,r2
i,α = (di, Pi), tl = ((tlλ1
, tlλ2
), tlµ).
Now, we shall introduce the ordering on the basis of the cell module
W [(s, (s1, s2)), ((λ1, λ2), µ)] as follows:
(Sr1,r2
i,α , tl) < (S
r′
1
,r′
2
j,β , tk)
(i) if (i, α, r1, r2) < (j, β, r′1, r
′
2) as in Definition 3.7 in PMK and
(ii) if (i, α, r1, r2) = (j, β, r′1, r
′
2) then (Sr1,r2
i,α , tl), (S
r′
1
,r′
2
j,β , tk) can be in-
dexed arbitrarily.
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250 Gram matrices and Stirling numbers
The above ordering can be used for the basis of the cell module
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)]
Arrange the basis of the cell module W [(s, (s1, s2)), ((λ1, λ2), µ)] and
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)] according to the order defined above and ob-
tain the Gram matrix G
λ,µ
2s1+s2
and
−→
G
λ,µ
2s1+s2
corresponding to the cell
modules W [(s, (s1, s2)), ((λ1, λ2), µ)] and
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)] re-
spectively.
Theorem 3.5. (i) The algebra of Z2-relations AZ2
k (x), signed partition
algebras
−→
AZ2
k (x) and partition algebras Ak(x) are semisimple over K(x)
where K is a field of characteristic zero where x is an indeterminate.
(ii) Suppose that the characteristic of the field K is 0, then
(a) the algebra of Z2-relations AZ2
k (q) is semisimple if and only if q
is not a root of the polynomial f(x) where f(x) =
∏
λ,µ
2k∏
2s1+s2=0
detGλ,µ
2s1+s2
where x = q and q ∈ C.
(b) the signed partition algebra
−→
AZ2
k (q) is semisimple if and only if q
is not a root of the polynomial f(x) where f(x) =
∏
λ,µ
2k∏
2s1+s2=0
det
−→
G
λ,µ
2s1+s2
.
(c) the partition algebra Ak(x) is semisimple if and only if q is not a
root of the polynomial f(x) where f(x) =
∏
λ
k∏
s=0
detGλ
s .
(iii) In particular,
(a) G
λ,µ
2s1+s2
coincides with Gk
2s1+s2
if
1. λ = ([s1],Φ) and µ = [s2] when s1, s2 6= 0,
2. λ = (Φ,Φ) and µ = [s2] when s1 = 0, s2 6= 0,
3. λ = ([s1],Φ) and µ = Φ when s1 6= 0, s2 = 0
4. λ = (Φ,Φ) and µ = Φ when s1, s2 = 0,
for 0 6 s1 6 k, 0 6 s2 6 k, 0 6 s1 + s2 6 k.
(b)
−→
G
λ,µ
2s1+s2
coincides with
−→
Gk
2s1+s2
if
1. λ = ([s1],Φ) and µ = [s2] when s1, s2 6= 0,
2. λ = (Φ,Φ) and µ = [s2] when s1 = 0, s2 6= 0,
3. λ = ([s1],Φ) and µ = Φ when s1 6= 0, s2 = 0
4. λ = (Φ,Φ) and µ = Φ when s1, s2 = 0,
for 0 6 s1 6 k − 1, 0 6 s2 6 k − 1, 0 6 s1 + s2 6 k − 1.
(c) Gλ
s coincides with Gk
s if
1. λ = s when s 6= 0,
2. λ = Φ when s = 0
for 0 6 s 6 k.
“adm-n2” — 2018/7/24 — 22:32 — page 251 — #89
N. Karimilla Bi, M. Parvathi 251
(iii)′ (a) If q is a root of the polynomial
f(x) =
2k∏
2s1+s2=0
detGk
2s1+s2
where detGk
2s1+s2
=
∏
06r16k−s1−s2
06r26k−s1−s2
2r1+r262k−2s1−2s2
det Ã2r1+r2,2r1+r2, Ã2r1+r2,2r1+r2 is as
in Theorem 2.14 then the algebra AZ2
k (q) is not semisimple.
In particular, q is an integer such that 0 6 q 6 k− 2 and q is a root of
the polynomial x2 − x− 2r′, 0 6 r′ 6 k − 2 then AZ2
k (q) is not semisimple.
(b) If q is a root of the polynomial
f(x) =
2k∏
2s1+s2=0
det
−→
Gk
2s1+s2
where det
−→
Gk
2s1+s2
=
∏
06r16k−s1−s2−1
06r26k−s1−s2−1
2r1+r262k−2s1−2s2−1
det
−̃→
A 2r1+r2,2r1+r2
∏
det
−̃→
A ρ,
−̃→
A 2r1+r2,2r1+r2 and
−̃→
A ρ are as in Theorem 2.14 then the algebra
−→
AZ2
k (q)
is not semisimple.
In particular, q is an integer such that 0 6 q 6 k− 2 and q is a root of
the polynomial x2 − x− 2r′, 0 6 r′ 6 k− 2 then
−→
AZ2
k (q) is not semisimple.
(c) If q is a root of the polynomial
f(x) =
k∏
s=0
detGk
s
where detGk
s =
∏
06r6k−s
det Ãr,r, Ãr,r is as in Theorem 2.14 then the
algebra Ak(q) is not semisimple.
(iv) The algebra of Z2-relations (AZ2
k (q)), signed partition algebra
(
−→
AZ2
k (q)) and the partition algebra (Ak(q)) over a field of characteristics
0 are quasi-hereditary for q 6= 0.
Proof. Part (i): The matrix of the bilinear form associated to the cell
module
−→
W [(s, (s1, s2)), ((λ1, λ2), µ)] as defined in Definition 4.3(ii) with
respect to the ordering of the basis as in Definition 3.4 is rewritten as
follows:
−→
G
λ,µ
2s1+s2
= (g(i,α,r1,r2),(j,β,r′1,r′2))16(i,α,r1,r2),(j,β,r′1,r
′
2
)6
−→
f 2s1+s2
“adm-n2” — 2018/7/24 — 22:32 — page 252 — #90
252 Gram matrices and Stirling numbers
where g(i,α,r1,r2),(j,β,r′1,r
′
2
) = a(i,α,r1,r2),(j,β,r′1,r
′
2
)B
λ,µ
δ1,δ2
,
a(i,α,r1,r2),(j,β,r′1,r
′
2
) =
xl(Pi∨Pj) if conditions (a) and (b)
of Definition 4.6 in [5] are satisfied,
0 otherwise,
B
λ,µ
δ1,δ2
= Bλ
δ1
⊗B
µ
δ2
with Bλ
δ1
= (φλ
δ1
(sλ, tλ)) and B
µ
δ2
= (φµ
δ2
(sµ, tµ)), B
λ
δ1
and B
µ
δ2
are the
matrices of the non-degenerate bilinear forms associated to the cell module
W λ and Wµ of the cellular algebras of K[Z2 ≀Ss1 ] and K[Ss2 ] respectively
as in Theorem 3.8 in [1] and δ1 and δ2 depends on the product of the
diagrams d
r1,r2
i,α and d
r′
1
,r′
2
j,β .
−→
G
λ,µ
2s1+s2
= (a(i,α,r1,r2),(j,β,r′1,r′2)B
λ,µ
δ1,δ2
)
16(i,α,r1,r2),(j,β,r′1,r
′
2
)6
−→
f 2s1+s2
The g(i,α,r1,r2),(i,α,r1,r2) = a(i,α,r1,r2),(i,α,r1,r2)A where A = B
λ,µ
1,1 =
Bλ
1 ⊗B
µ
1 . Thus, the leading coefficient of the Gram matrix is
(detA)
−→
f 2s1+s2
×dim
−→
W [(s,(s1,s2)),((λ1,λ2),µ)]
which is non-zero over a characteristic zero. Therefore, the algebra
−→
AZ2
k (x)
is semisimple. The proof for the algebra of Z2-relations and the partition
algebras are similar to the above proof.
Part (ii): By Theorem 3.8 in [1],
−→
AZ2
k is semisimple if and only if
detGλ,µ
2s1+s2
6= 0 for all s1, s2 and for all λ, µ, since
detGλ,µ
2s1+s2
6= 0 if and only if Φ is non-degenerate.
Part (iii)(b): Now,
−→
G
λ,µ
2s1+s2
=
−→
Gk
2s1+s2
if
1) λ = ([s1],Φ) and µ = [s2] when s1, s2 6= 0,
2) λ = (Φ,Φ) and µ = [s2] when s1 = 0, s2 6= 0,
3) λ = ([s1],Φ) and µ = Φ when s1 6= 0, s2 = 0
for 0 6 s1 6 k − 1, 0 6 s2 6 k − 1, 0 6 s1 + s2 6 k − 1, since A is the
1× 1 identity matrix,
If λ = (Φ,Φ) and µ = Φ when s1, s2 = 0, then
−→
G
Φ,Φ
2s1+s2
coincides with
−→
Gk
0+0.
“adm-n2” — 2018/7/24 — 22:32 — page 253 — #91
N. Karimilla Bi, M. Parvathi 253
Part (iii)′(b): If q is a root of
f(x) =
∏
06r16k−s1−s2−1
06r26k−s1−s2−1
2r1+r262k−2s1−2s2−1
det
−̃→
A 2r1+r2,2r1+r2
∏
det
−̃→
A ρ,
then det
−→
Gk
2s1+s2
= 0 = det
−→
G
(([s1],Φ),[s2])
2s1+s2
. Thus, the algebra
−→
AZ2
k is not
semisimple.
In particular, by Remark 2.15 if q is an integer such that 0 6 q 6 k−2
and q is a root of polynomial x2 − x− 2r′, 0 6 r′ 6 k− 2 then the algebra
−→
AZ2
k is not semisimple.
The proof of (a) and (c) is similar to the proof of (b).
Part (iv): It follows from Remark 3.10 in [1] and Theorem 5.4 in [5].
Appendix
The following is an example of Gram matrix in
−→
AZ2
3 (x).
Let s1 = 1 and s2 = 0. The following are the diagrams in J6
2×1+0.
d
0,0
1,α1
= d
0,0
2,α1
= d
0,0
3,α1
= d
0,0
4,α1
= d
0,1
5,α2
=
d
0,1
6,α2
= d
0,1
7,α2
= d
0,1
8,α2
= d
0,1
9,α2
= d
0,1
10,α2
=
d
0,1
11,α3
= d
0,1
12,α3
= d
0,1
13,α3
= d
1,0
14,α4
= d
1,0
15,α4
=
d
1,0
21,α5
= d
1,0
22,α5
= d
1,0
23,α5
= d
1,0
24,α5
= d
1,0
25,α5
=
d
1,0
16,α4
= d
1,0
17,α4
= d
1,0
18,α4
= d
1,0
19,α4
= d
1,0
20,α5
=
d
1,1
26,α6
= d
1,1
27,α6
= d
1,1
28,α6
= d
1,1
29,α6
= d
1,1
30,α6
=
d
1,1
31,α6
= d
2,0
32,α7
= d
2,0
33,α7
= d
2,0
34,α7
=
where α1 = (3,Φ,Φ,Φ), α2 = (2,Φ,Φ, 1), α3=(1,Φ,Φ, 2), α4=(2,Φ, 1,Φ),
α5 = (1,Φ, 0, 2), α6 = (1,Φ, 1, 1), α7 = (1, 0, 12, 0) and d
r1,r2
i,α is a dia-
gram having r1 number of pairs of {e}-horizontal edges, r2 number of
Z2-horizontal edges and α is the underlying partition of dr1,r2i,α .
“adm-n2” — 2018/7/24 — 22:32 — page 254 — #92
254 Gram matrices and Stirling numbers
d
0
,0
1
,α
1
d
0
,0
2
,α
1
d
0
,0
3
,α
1
d
0
,0
4
,α
1
d
0
,1
5
,α
2
d
0
,1
6
,α
2
d
0
,1
7
,α
2
d
0
,1
8
,α
2
d
0
,1
9
,α
2
d
0
,1
1
0
,α
2
d
0
,1
1
1
,α
3
d
0
,1
1
2
,α
3
d
0
,1
1
3
,α
3
d
1
,0
1
4
,α
4
d
1
,0
1
5
,α
4
d
1
,0
1
6
,α
4
d
1
,0
1
7
,α
4
d
1
,0
1
8
,α
4
d
1
,0
1
9
,α
4
d
1
,0
2
0
,α
5
d
1
,0
2
1
,α
5
d
1
,0
2
2
,α
5
d
1
,0
2
3
,α
5
d
1
,0
2
4
,α
5
d
1
,0
2
5
,α
5
d
1
,1
2
6
,α
6
d
1
,1
2
7
,α
6
d
1
,1
2
8
,α
6
d
1
,1
2
9
,α
6
d
1
,1
3
0
,α
6
d
1
,1
3
1
,α
6
d
2
,0
3
2
,α
7
d
2
,0
3
3
,α
7
d
2
,0
3
4
,α
7
d
0
,0
1
,α
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
1
1
1
d
0
,0
2
,α
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
1
0
1
0
1
1
0
0
0
0
0
0
0
1
1
1
d
0
,0
3
,α
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
0
1
0
1
0
0
0
0
0
0
1
1
1
d
0
,0
4
,α
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
1
1
0
0
1
0
0
0
0
0
0
1
1
1
d
0
,1
5
,α
2
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
0
x
0
x
0
0
x
x
0
d
0
,1
6
,α
2
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
0
x
0
x
0
0
x
x
0
d
0
,1
7
,α
2
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
x
0
0
0
0
x
x
0
x
d
0
,1
8
,α
2
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
x
x
0
x
d
0
,1
9
,α
2
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
x
0
x
0
0
x
x
d
0
,1
1
0
,α
2
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
0
x
0
0
x
x
d
0
,1
1
1
,α
3
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
x
x
0
0
0
0
x
x
0
0
0
0
x
0
0
d
0
,1
1
2
,α
3
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
x
x
0
0
0
0
x
x
0
0
0
x
0
d
0
,1
1
3
,α
3
0
0
0
0
0
0
0
0
0
0
0
0
x
0
0
0
0
0
0
0
0
0
0
x
x
0
0
0
0
x
x
0
0
x
d
1
,0
1
4
,α
4
1
1
0
0
x
0
0
0
0
0
0
0
0
x
2
0
1
1
1
1
1
1
1
1
0
0
0
x
0
x
x
0
x
2
x
2
0
d
1
,0
1
5
,α
4
0
0
1
1
0
x
0
0
0
0
0
0
0
0
x
2
1
1
1
1
1
1
1
1
0
0
0
x
0
x
x
0
x
2
x
2
0
d
1
,0
1
6
,α
4
1
0
0
1
0
0
x
0
0
0
0
0
0
1
1
x
2
0
1
1
1
1
0
0
1
1
x
0
0
0
0
x
x
2
0
x
2
d
1
,0
1
7
,α
4
0
1
1
0
0
0
0
x
0
0
0
0
0
1
1
0
x
2
1
1
1
1
0
0
1
1
x
0
0
0
0
x
x
2
0
x
2
d
1
,0
1
8
,α
4
1
0
1
0
0
0
0
0
x
0
0
0
0
1
1
1
1
x
2
0
0
0
1
1
1
1
0
0
x
0
x
0
0
x
2
x
2
d
1
,0
1
9
,α
4
0
1
0
1
0
0
0
0
0
x
0
0
0
1
1
1
1
0
x
2
0
0
1
1
1
1
0
0
x
0
x
0
0
x
2
x
2
d
1
,0
2
0
,α
5
1
0
1
0
0
0
0
0
0
0
x
0
0
1
1
1
1
0
0
x
2
x
1
1
1
1
x
x
0
0
0
0
x
2
0
0
d
1
,0
2
1
,α
5
0
1
0
1
0
0
0
0
0
0
x
0
0
1
1
1
1
0
0
x
x
2
1
1
1
1
x
x
0
0
0
0
x
2
0
0
d
1
,0
2
2
,α
5
1
0
0
1
0
0
0
0
0
0
0
x
0
1
1
0
0
1
1
1
1
x
2
x
1
1
0
0
x
x
0
0
0
x
2
0
d
1
,0
2
3
,α
5
0
1
1
0
0
0
0
0
0
0
0
x
0
1
1
0
0
1
1
1
1
x
x
2
1
1
0
0
x
x
0
0
0
x
2
0
d
1
,0
2
4
,α
5
1
1
0
0
0
0
0
0
0
0
0
0
x
0
0
1
1
1
1
1
1
1
1
x
2
x
0
0
0
0
x
x
0
0
x
2
d
1
,0
2
5
,α
5
0
0
1
1
0
0
0
0
0
0
0
0
x
0
0
1
1
1
1
1
1
1
1
x
x
2
0
0
0
0
x
x
0
0
x
2
d
1
,1
2
6
,α
6
0
0
0
0
0
0
x
x
0
0
x
0
0
0
0
x
x
0
0
x
x
0
0
0
0
x
3
x
2
0
0
0
0
x
3
0
0
d
1
,1
2
7
,α
6
0
0
0
0
x
x
0
0
0
0
x
0
0
x
x
0
0
0
0
x
x
0
0
0
0
x
2
x
3
0
0
0
0
x
3
0
0
d
1
,1
2
8
,α
6
0
0
0
0
0
0
0
0
x
x
0
x
0
x
x
0
0
x
x
0
0
x
x
0
0
0
0
x
3
x
2
0
0
0
x
3
0
d
1
,1
2
9
,α
6
0
0
0
0
x
x
0
0
0
0
0
x
0
x
x
0
0
0
0
0
0
x
x
0
0
0
0
x
2
x
3
0
0
0
x
3
0
d
1
,1
3
0
,α
6
0
0
0
0
0
0
0
0
x
x
0
0
x
0
0
0
0
x
x
0
0
0
0
x
x
0
0
0
0
x
3
x
2
0
0
x
3
d
1
,1
3
1
,α
6
0
0
0
0
0
0
x
x
0
0
0
0
x
0
0
x
x
0
0
0
0
0
0
x
x
0
0
0
0
x
2
x
3
0
0
x
3
d
2
,0
3
2
,α
7
1
1
1
1
x
x
x
x
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x
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2
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2
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0
0
d
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3
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7
1
1
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4
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7
1
1
1
1
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x
x
x
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x
0
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x
2
x
2
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4
“adm-n2” — 2018/7/24 — 22:32 — page 255 — #93
N. Karimilla Bi, M. Parvathi 255
After applying the column operations and by Theorem 2.14 the matrix
−→
G3
2.1+0 reduces as follows:
−→
A 0,0 ∼
−̃→
A 0,0 = I4,
−→
A 1,1 ∼
−̃→
A 1,1 = xI9,
−→
A 2,2 ∼
−̃→
A 2,2 = (x2 − x− 2)I12 + (−2)I ′12.
where In denotes n× n identity matrix and I ′n denotes n× n off-diagonal
matrix.
After applying the row and column operations, the matrix
−→
A ρ is
reduced as follows:
−→
A ρ ∼
d
1,1
26,α6
d
1,1
27,α6
d
1,1
28,α6
d
1,1
29,α6
d
1,1
30,α6
d
1,1
31,α6
d
2,0
32,α7
d
2,0
33,α7
d
2,0
34,α7
d
1,1
26,α6
x3
−3x x2
−x 0 0 0 −2x −x2
+x 0 0
d
1,1
27,α6
x2
−x x3
−3x 0 −2x 0 0 −x2
+x 0 0
d
1,1
28,α6
0 0 x3
−3x x2
−x −2x 0 0 −x2
+x 0
d
1,1
29,α6
0 −2x x2
−x x3
−3x 0 0 0 −x2
+x 0
d
1,1
30,α6
0 0 −2x 0 x3
−3x x2
−x 0 0 −x2
+x
d
1,1
31,α6
−2x 0 0 0 x2
−x x3
−3x 0 0 −x2
+x
d
2,0
32,α7
−x2
+x −x2
+x 0 0 0 0
x
4
−2x
3
−4x
2
+5x+8
−2x2
+2x+8 −2x2
+2x+8
d
2,0
33,α7
0 0 −x2
+x −x2
+x 0 0 −2x2
+2x+8
x
4
−2x
3
−4x
2
+5x+8
−2x2
+2x+8
d
2,0
34,α7
0 0 0 0 −x2
+x −x2
+x −2x2
+2x+8 −2x2
+2x+8
x
4
−2x
3
−4x
2
+5x+8
The entry x2 − x in the above matrix cannot be eliminated while
applying column operations since the following diagrams do not belong to
−→
AZ2
3 (x).
, ,
Acknowledgement
The authors would like to express their gratitude and sincere thanks
to the referee for all his(her) valuable comments and suggestions which in
turn made the paper easy to read.
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Contact information
N. Karimilla Bi,
M. Parvathi
Ramanujan Institute for Advanced Study in
Mathematics, University of Madras, Chepauk,
Chennai -600 005, Tamilnadu, India
E-Mail(s): sparvathi@hotmail.com
Received by the editors: 22.09.2015
and in final form 16.03.2018.
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