Hyperdeterminantal Total Positivity
For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal...
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| author | Johnson, Kenneth W. Richards, Donald St. P. |
| author_facet | Johnson, Kenneth W. Richards, Donald St. P. |
| citation_txt | Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages |
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| description | For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel (₁, …, ₂ₘ) = exp(₁⋯₂ₘ), ₁,…,₂ₘ ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula. We use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity via the theory of finite reflection groups are described, and some open problems are posed.
|
| first_indexed | 2026-03-20T16:46:49Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 055, 18 pages
Hyperdeterminantal Total Positivity
Kenneth W. JOHNSON a and Donald St. P. RICHARDS b
a) Department of Mathematics, Pennsylvania State University,
Abington, Pennsylvania 19001, USA
E-mail: kwj1@psu.edu
b) Department of Statistics, Pennsylvania State University,
University Park, PA 16802, USA
E-mail: dsr11@psu.edu
Received December 04, 2024, in final form July 03, 2025; Published online July 11, 2025
https://doi.org/10.3842/SIGMA.2025.055
Abstract. For a given positive integer m, the concept of hyperdeterminantal total pos-
itivity is defined for a kernel K : R2m → R, thereby generalizing the classical concept of
total positivity. Extending the fundamental example, K(x, y) = exp(xy), x, y ∈ R, of
a classical totally positive kernel, the hyperdeterminantal total positivity property of the
kernel K(x1, . . . , x2m) = exp(x1 · · ·x2m), x1, . . . , x2m ∈ R is established. By applying Mat-
sumoto’s hyperdeterminantal Binet–Cauchy formula, we derive a generalization of Karlin’s
basic composition formula; then we use the generalized composition formula to construct
several examples of hyperdeterminantal totally positive kernels. Further generalizations of
hyperdeterminantal total positivity by means of the theory of finite reflection groups are
described and some open problems are posed.
Key words: Binet–Cauchy formula; determinant; generalized hypergeometric functions of
matrix argument; Haar measure; hyperdeterminant; Schur function; unitary group; zonal
polynomials
2020 Mathematics Subject Classification: 33C20; 05E05; 15A15; 15A72; 33C80
This article is dedicated to Steve Milne, who has
displayed over the years a masterful use of the
Schur functions in algebraic and analytic set-
tings. We heartily congratulate Steve on the oc-
casion of his 75th birthday and wish him many
more years of beautiful applications of the Schur
functions and their properties.
1 Introduction
The theory of total positivity has, for the past 95 years, played an increasingly important role
in many aspects of the mathematical sciences. The theory was largely initiated by the work
of Schoenberg [35] and Krein [25] in the 1930’s, Pólya [33] in the 1940’s, and Gantmacher and
Krein [9] and Karlin [21] in the 1950’s, and it is remarkable that each of those early authors was
motivated by a wide variety of considerations such as approximation theory, frequency sequences
and functions, integral operators, mathematical economics, ordinary differential equations and
related Green’s functions, reliability theory, splines, statistics, and variation-diminishing trans-
formations.
This paper is a contribution to the Special Issue on Basic Hypergeometric Series Associated with Root
Systems and Applications in honor of Stephen C. Milne’s 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Milne.html
mailto:kwj1@psu.edu
mailto:dsr11@psu.edu
https://doi.org/10.3842/SIGMA.2025.055
https://www.emis.de/journals/SIGMA/Milne.html
2 K.W. Johnson and D.St.P. Richards
Since the appearance of Karlin’s monograph [21], the theory of total positivity has broadened
immensely, to the point where it arises in combinatorics, classical special functions, Gröbner
bases, matrix analysis, Lie groups, harmonic analysis, special functions of matrix argument,
mathematical physics, operations research, partial differential equations, stochastic processes,
actuarial mathematics, conformal field theory (high energy physics), statistical mechanics, evo-
lutionary biology, computer-aided geometric design, and many other areas. We refer to Gant-
macher and Krein [9], Gasca and Micchelli [10], Karlin [21, 22], Pinkus [32], and Gelfand,
Kapranov, and Zelevinsky [13] for various accounts and applications of the classical theory and
numerous references to the larger literature.
Let D ⊆ R2 and d ∈ N. In the classical setting, a kernel K : D → R is totally positive of
order d, denoted TPd, if the n× n determinant
det(K(xj , yk)) =
∣∣∣∣∣∣∣∣∣
K(x1, y1) K(x1, y2) · · · K(x1, yn)
K(x2, y1) K(x2, y2) · · · K(x2, yn)
...
...
...
K(xn, y1) K(xn, y2) · · · K(xn, yn)
∣∣∣∣∣∣∣∣∣ (1.1)
is nonnegative for all n = 1, . . . , d, and for all x1 > · · · > xd and y1 > · · · > yd such
that (xj , yk) ∈ D for all j, k = 1, . . . , d. Equivalently, for a kernel K : D → [0,∞), total posi-
tivity of order d means that all minors of the d × d matrix (K(xi, yj)) are nonnegative when-
ever x1 > · · · > xd and y1 > · · · > yd, i.e., all n × n submatrices of the matrix (K(xi, yj)) have
nonnegative determinant, for all n = 1, . . . , d.
Consider the kernel
K(x, y) = exp(xy), (x, y) ∈ R2, (1.2)
a function that is the fundamental example of a totally positive kernel. This kernel will provide
a guiding role in our subsequent results. It is well known that this kernel is totally positive of
order d for all d ∈ N, a property that is denoted by TP∞. Moreover, the minor (1.1) is positive
for all n ∈ N, so we say that K is strictly totally positive of order infinity, denoted STP∞.
Returning to the general definition in (1.1), it may be observed that the classical concept of
total positivity is defined (only) for kernels defined on R2 and for all orders d ∈ N. On the other
hand, for kernels K : Rm → R, where m ≥ 2, the concept of total positivity is defined only for
orders d ≤ 2, with d = 1 signifying thatK is nonnegative; specifically, the concept ofmultivariate
totally positivity of order 2 for kernels K : Rm → [0,∞) was defined in the article [23]. Thus
no general treatment has yet been developed for the concept of multivariate total positivity of
order d, for any d > 2, when the kernel K is defined on Rm, for arbitrary m > 2.
In this article, for a given positive integer m, we follow [20, 38] by applying the theory
of Cayley’s first hyperdeterminant to define the concept of hyperdeterminantal total positivity
(HTP) for a kernel K : R2m → R. For the case in which m = 1, this definition reduces to the
classical concept of total positivity which was defined through the minors in (1.1). Generalizing
the fundamental example in (1.2) of a classical totally positive kernel, we shall establish that
the kernel
K(x1, . . . , x2m) = exp(x1 · · ·x2m), (x1, . . . , x2m) ∈ R2m, (1.3)
is a HTP kernel of arbitrarily large order. We show, moreover, that the resulting hyperdeter-
minants are strictly positive, which generalizes the STP∞ property of the fundamental exam-
ple (1.2).
We remark that the articles [38] derived results for the total positivity of Hankel and Van-
dermonde arrays that are equivalent to the hyperdeterminantal properties of those kernels.
Hyperdeterminantal Total Positivity 3
However, to the best of our knowledge, our results in the present article appear to be the first
to connect the theory of HTP arrays with the generalized hypergeometric functions of matrix
argument.
In Section 2, we provide the definition of the hyperdeterminant and several of its properties.
We provide in Section 3 a hyperdeterminantal Binet–Cauchy formula, due to Matsumoto [30],
and then we apply that result to obtain a Schur function summation formula for the hyperde-
terminant constructed from the kernel (1.3).
In Section 4, we extend to the hyperdeterminantal setting an integral of Harish-Chandra [17]
that now appears prominently in the theory of total positivity [15, 16]. Further, we apply the
extended integral to derive Schur function summation formulas for certain hyperdeterminants
defined in terms of the classical generalized hypergeometric series. In Section 5, we provide the
definition of hyperdeterminantal total positivity, obtain several examples of such kernels, and
generalize the classical basic composition formula [21, p. 17]. As consequences of the hyperdeter-
minantal Binet–Cauchy formula and the generalized basic composition formula, we demonstrate
how numerous examples of HTP kernels can be constructed.
Finally, in Section 6, we describe some directions for future research that are opened by this
article. In particular, we raise the possibility of investigating further generalizations of HTP
kernels by means of the theory of finite reflection groups, and we raise the problem of deriving
hyperdeterminantal generalizations of the FKG inequality.
2 Hyperdeterminants
For any collection of indices r1, . . . , rl ∈ {1, . . . , n}, let A(r1, . . . , rl) ∈ C and form the multidi-
mensional array, or tensor, A = (A(r1, . . . , rl))1≤r1,...,rl≤n. Also let Sn denote the symmetric
group on n symbols. Cayley’s first definition in [4, 5, 6] of the hyperdeterminant of the array A is
Det
(
A(r1, . . . , rl)
)
1≤r1,...,rl≤n
=
1
n!
∑
σ1∈Sn
· · ·
∑
σl∈Sn
(
l∏
k=1
sgn(σk)
)
·
n∏
j=1
A(σ1(j), . . . , σl(j)).
By replacing each σr in the multiple summation by σ0σr, where σ0 is an odd permutation, we
find that Det
(
A(r1, . . . , rl)
)
= (−1)l Det
(
A(r1, . . . , rl)
)
. Consequently, Det
(
A(r1, . . . , rl)
)
≡ 0
if l is odd, a well known result; see, e.g., [28, Section 2]. Therefore, we assume henceforth that
l is even, l = 2m, so that
Det(A(r1, . . . , r2m))1≤r1,...,r2m≤n
:=
1
n!
∑
σ1∈Sn
· · ·
∑
σ2m∈Sn
(
2m∏
k=1
sgn(σk)
)
·
n∏
j=1
A(σ1(j), . . . , σ2m(j)). (2.1)
For simplicity, we will denote Det(A(r1, . . . , r2m))1≤r1,...,r2m≤n by Det(A) if there is no possibility
of confusion.
The hyperdeterminant satisfies many properties that generalize the properties of the classical
determinant. For a wide range of such properties, and numerous applications, we refer to [4,
5, 6, 8, 12, 13, 19, 26, 31, 34, 36, 37]. For applications to mathematical physics and to the
calculation of Selberg’s famous integral, we refer to [3, 27, 28].
Let us express the formula (2.1) in two ways. First, observe that the sum over σ2m is a classical
determinant: For fixed permutations σ1, . . . , σ2m−1 ∈ Sn,
∑
σ2m∈Sn
sgn(σ2m)
n∏
j=1
A(σ1(j), . . . , σ2m(j))
4 K.W. Johnson and D.St.P. Richards
≡ det(A(σ1(i), . . . , σ2m−1(i), j))1≤i,j≤n
= sgn(σ2m−1) det
(
A
(
σ1σ
−1
2m−1(i), . . . , σ2m−2σ
−1
2m−1(i), i, j
))
1≤i,j≤2
.
Therefore,
n! Det(A) =
∑
σ1,...,σ2m−1∈Sn
(
2m−2∏
k=1
sgn(σk)
)
× det
(
A(σ1σ
−1
2m−1(i), . . . , σ2m−2σ
−1
2m−1(i), i, j)
)
1≤i,j≤n
.
Replacing σk by σkσ2m−1, 1 ≤ k ≤ 2m− 2, we obtain
Det(A) =
∑
σ1,...,σ2m−2∈Sn
(
2m−2∏
k=1
sgn(σk)
)
· det(A(σ1(i), . . . , σ2m−2(i), i, j))1≤i,j≤n. (2.2)
This represents Det(A) as an alternating multisum of classical determinants.
The second way to express the hyperdeterminant is by rewriting (2.1) as
n! Det(A) =
∑
σ2m−1,σ2m∈Sn
2m∏
k=2m−1
sgn(σk)
×
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
A(σ1(j), . . . , σ2m−2(j), σ2m−1(j), σ2m(j)).
For fixed r2m−1, r2m ∈ {1, . . . , n}, define the multidimensional array Br2m−1,r2m(r1, . . . , r2m−2) =
A(r1, . . . , r2m−2, r2m−1, r2m), where r1, . . . , r2m−2 ∈ {1, . . . , n}. Then, for fixed σ2m−1, σ2m ∈ Sn,
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
A(σ1(j), . . . , σ2m−2(j), σ2m−1(j), σ2m(j))
=
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
Bσ2m−1(j),σ2m(j)(σ1(j), . . . , σ2m−2(j))
=
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
Bσ2m−1σ
−1
2m(j),j
(
σ1σ
−1
2m(j), . . . , σ2m−2σ
−1
2m(j)
)
=
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
Bσ2m−1σ
−1
2m(j),j(σ1(j), . . . , σ2m−2(j)).
Therefore,
Det(A) =
1
n!
∑
σ2m−1,σ2m∈Sn
2m∏
k=2m−1
sgn(σk)
×
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
Bσ2m−1σ
−1
2m(j),j(σ1(j), . . . , σ2m−2(j)).
Hyperdeterminantal Total Positivity 5
On replacing σ2m−1 by σ2m−1σ2m, we obtain
Det(A) =
∑
σ2m−1∈Sn
sgn(σ2m−1)
×
∑
σ1,...,σ2m−2∈Sn
2m−2∏
k=1
sgn(σk) ·
n∏
j=1
Bσ2m−1(j),j(σ1(j), . . . , σ2m−2(j))
= n!
∑
σ2m−1∈Sn
sgn(σ2m−1)Det(Bσ2m−1(j),j(σ1(j), . . . , σ2m−2(j)). (2.3)
This represents Det(A) as a single alternating sum of hyperdeterminants.
3 A Binet–Cauchy theorem for hyperdeterminants
In reviewing the hyperdeterminantal literature, one observes that hyperdeterminants satisfy
various generalizations of the classical Binet–Cauchy formula. Variants of the Binet–Cauchy
formula for multidimensional arrays have appeared in the literature going back to [11, 19], and
other analogs of the formula have appeared in [2, 3, 7, 8, 24, 30]. In all of those cited articles,
the proof of the corresponding generalized Binet–Cauchy formula is analogous to the proof of
the classical Binet–Cauchy formula [21] and also to generalizations of the formula that are based
on the theory of finite reflection groups [15, 16].
The statement and proof of the following generalized Binet–Cauchy formula are due to Mat-
sumoto [30, Proposition 2.1].
Proposition 3.1 (Matsumoto [30]). Let (M, µ) be a sigma-finite measure space and let {ϕk,r |
1 ≤ k ≤ 2m, 1 ≤ r ≤ n} be a collection of complex-valued functions on M such that, for
all r1, . . . , r2m satisfying 1 ≤ r1, . . . , r2m ≤ n, the integral
A(r1, . . . , r2m) :=
∫
M
2m∏
k=1
ϕk,rk(x)dµ(x), (3.1)
converges absolutely. Then, with Mn := M× · · · ×M (n factors),
Det(A(r1, . . . , r2m)) =
1
n!
∫
Mn
2m∏
k=1
det(ϕk,r(xs))1≤r,s≤n ·
n∏
j=1
dµ(xj). (3.2)
Moreover, if M is totally ordered then
Det(A(r1, . . . , r2m)) =
∫
· · ·
∫
x1>···>xn
x1,...,xn∈M
2m∏
k=1
det(ϕk,r(xs))1≤r,s≤n ·
n∏
j=1
dµ(xj). (3.3)
Proof. By (2.1),
n! Det(A(r1, . . . , r2m)) =
∑
σ1∈Sn
· · ·
∑
σ2m∈Sn
(
2m∏
k=1
sgn(σk)
)
n∏
j=1
(∫
M
2m∏
k=1
ϕk,σk(j)(x)dµ(x)
)
=
∫
Mn
∑
σ1∈Sn
· · ·
∑
σ2m∈Sn
2m∏
k=1
(
sgn(σk)
n∏
j=1
ϕk,σk(j)(xj)
)
n∏
j=1
dµ(xj).
6 K.W. Johnson and D.St.P. Richards
On writing the multi-sum over σ1, . . . , σ2m as a product of summations, we obtain
n! Det(A(r1, . . . , r2m)) =
∫
Mn
2m∏
k=1
( ∑
σk∈Sn
sgn(σk)
n∏
j=1
ϕk,σk(j)(xj)
)
n∏
j=1
dµ(xj)
=
∫
Mn
2m∏
k=1
det(ϕk,r(xs))1≤r,s≤n
n∏
j=1
dµ(xj), (3.4)
and this establishes (3.2).
Consider next the case in which M is totally ordered. On observing that all k determinants
in the integrand in (3.4) are identically zero if any two xs coincide, it follows that the integral
in (3.4) reduces to an integral over the union of disjoint open Weyl chambers,⋃
σ∈Sn
{(x1, . . . , xn) ∈ Mn | xσ(1) > · · · > xσ(n)}.
For each σ ∈ Sn, consider the corresponding Weyl chamber
Cn(σ) := {(x1, . . . , xn) ∈ Mn | xσ(1) > · · · > xσ(n)};
then
n! Det(A(r1, . . . , r2m)) =
∑
σ∈Sn
∫
Cn(σ)
2m∏
k=1
det(ϕk,r(xs))1≤r,s≤n
n∏
j=1
dµ(xj).
On the chamber Cn(σ) we make the change-of-variables (xσ(1), . . . , xσ(n)) → (x1, . . . , xn), thereby
obtaining∫
Cn(σ)
2m∏
k=1
det(ϕk,r(xs))1≤r,s≤n
n∏
j=1
dµ(xj) =
∫
Cn
2m∏
k=1
(sgn(σ) det(ϕk,r(xs))1≤r,s≤n)
n∏
j=1
dµ(xj),
where Cn = {(x1, . . . , xn) ∈ Mn | x1 > · · · > xn} is the fundamental Weyl chamber. Noting that
there are n! chambers, each of which corresponds to a unique permutation σ ∈ Sn, and also
that (sgn(σ))2m ≡ 1, we obtain (3.3). ■
We now give an application of Proposition 3.1 that will be crucial for establishing later the
hyperdeterminantal total positivity properties of the kernel (1.3).
Example 3.2. In Proposition 3.1 set M = {0, 1, 2, . . . }, the set of nonnegative integers. Let µ
be the discrete (counting) measure with weight µ({i}) = 1/i!, i ∈ M. For 1 ≤ k ≤ 2m,
1 ≤ r ≤ n, and i ∈ M, let xk = (xk,1, . . . , xk,n) ∈ Rn and define ϕk,r(i) = xik,r. By (3.1),
A(r1, . . . , r2m) =
∞∑
i=0
xi1,r1 · · ·x
i
2m,r2m
i!
= exp(x1,r1 · · ·x2m,r2m).
On applying (3.3), we obtain the hyperdeterminantal summation formula
Det(exp(x1,r1 · · ·x2m,r2m)) =
∑
i1>···>in≥0
1
i1! · · · in!
·
2m∏
k=1
det
(
xisk,r
)
1≤r,s≤n
. (3.5)
Define λs = is − n + s, s = 1, . . . , n; then λ1 ≥ · · · ≥ λn ≥ 0, so the vector λ = (λ1, . . . , λn)
is a partition. Also let
V (xk) =
∏
1≤r<s≤n
(xk,r − xk,s), (3.6)
Hyperdeterminantal Total Positivity 7
be the Vandermonde determinant in the variables xk = (xk,1, . . . , xk,n); then
sλ(xk) =
det
(
xisk,r
)
1≤r,s≤n
V (xk)
(3.7)
is the well-known Schur function [29, p. 40]. Writing (3.7) in the form
det
(
xisk,r
)
1≤r,s≤n
= V (xk)sλ(xk), (3.8)
and applying this to (3.5), we obtain a summation formula that expresses the corresponding
hyperdeterminant in terms of a sum of weighted products of Schur functions
Det(exp(x1,r1 · · ·x2m,r2m))∏2m
k=1 V (xk)
=
∑
λ
n∏
j=1
1
(λj + n− j)!
·
2m∏
k=1
sλ(xk), (3.9)
where the sum is taken over all partitions λ = (λ1, . . . , λn) of all nonnegative integers such that λ
is of length ℓ(λ) ≤ n.
4 A hyperdeterminantal extension
of an integral of Harish-Chandra
In this section, we extend an integral of Harish-Chandra [17] that has played a prominent
role in the theory of total positivity [15, 16]. Further, we apply the extended integral to derive
Schur function summation formulas for some hyperdeterminants defined in terms of the classical
generalized hypergeometric series, and this will also connect the theory of hyperdeterminatal
total positivity with the hypergeometric functions of Hermitian matrix argument.
Denote by Hn×n the space of n × n Hermitian matrices. In the sequel, if X ∈ Hn×n has
eigenvalues x1, . . . , xn and λ is a partition then we will write sλ(X) for sλ(x1, . . . , xn), the
Schur function with arguments x1, . . . , xn. Since every Hermitian matrix can be diagonalized
by a unitary transformation, then sλ(X) is a unitarily invariant polynomial function on Hn×n.
For X1, X2 ∈ Hn×n, we write sλ(X1X2) to denote sλ(w1, . . . , wn), where w1, . . . , wn are
the eigenvalues of X1X2. Since the matrices X1X2 and X2X1 have the same eigenvalues,
then sλ(X1X2) = sλ(X2X1).
For the case in which m = 1, the summation formula (3.9) reduces to a sum that arose
crucially in [15], having been used there to derive a new proof that the fundamental clas-
sical example, K(x, y) = exp(xy), (x, y) ∈ R2, is strictly totally positive of order infinity
(STP∞). Let βn =
∏n
j=1(j − 1)! and denote by U(n) the group of n × n unitary matrices.
For generic U ∈ U(n) let dU be the Haar measure on U(n), normalized to have total volume 1.
As noted in [15], the special case with m = 1 of (3.9) also appears prominently in an integral
formula of Harish-Chandra [17]:
Proposition 4.1 (Harish-Chandra [17]). Suppose that X1, X2 ∈ Hn×n have eigenvalues x1,1,
x1,2, . . . , x1,n and x2,1, x2,2, . . . , x2,n, respectively. Then
det(exp(x1,r1x2,r2))1≤r1,r2≤n
V (x1)V (x2)
= β−1
n
∫
U(n)
exp
(
trUX1U
−1X2
)
dU. (4.1)
Setting m = 1 in (3.9), and then comparing the right-hand sides of (4.1) and (3.9), it follows
that ∫
U(n)
exp
(
trUX1U
−1X2
)
dU = βn
∑
λ
n∏
j=1
1
(λj + n− j)!
· sλ(x1)sλ(x2). (4.2)
8 K.W. Johnson and D.St.P. Richards
As it turns out, an alternative proof of (4.2) was obtained in [15] by means of the theory
of zonal polynomials and the hypergeometric functions of matrix argument. To extend the
formula (4.2) to the setting of hyperdeterminants, we will again apply the theory of zonal
polynomials. In the proof of the next result, we will largely retain the notation of [15, Section 2]
and will follow closely the exposition given there. In particular, if λ = (λ1, . . . , λn) is a partition
of length ℓ(λ) ≤ n then the weight of λ is |λ| = λ1 + · · ·+ λn.
For a ∈ C and j = 0, 1, 2, . . . , the classical rising factorial is defined as
(a)j =
Γ(a+ j)
Γ(a)
= a(a+ 1) · · · (a+ j − 1). (4.3)
For a ∈ C and any partition λ of length ℓ(λ) ≤ n, the partitional rising factorial is
(a)λ =
n∏
j=1
(a− j + 1)λj
. (4.4)
Define
ωλ = |λ|!
∏
1≤i<j≤n(λi − λj − i+ j)∏n
j=1(λj + n− j)!
and set
dλ := sλ(In) =
∏
1≤i<j≤n(λi − λj − i+ j)∏n
j=1(j − 1)!
.
Then it is straightforward to see that
ωλ =
|λ|!dλ
(n)λ
. (4.5)
Further, for any n × n Hermitian matrix X and partition λ, define the zonal polynomial as
normalized Schur function,
Zλ(X) = ωλsλ(X). (4.6)
As shown in [15, Section 2], it is a consequence of (4.4)–(4.5) that, for all l = 0, 1, 2, . . . , the
normalization in (4.6) leads to the expansion
(trX)l =
∑
|λ|=l
Zλ(X). (4.7)
The zonal polynomials Zλ satisfy the mean-value property: For any n × n Hermitian matri-
ces X1 and X2,∫
U(n)
Zλ
(
UX1U
−1X2
)
dU =
Zλ(X1)Zλ(X2)
Zλ(In)
. (4.8)
The mean-value property also implies that∫
U(n)
∫
U(n)
Zλ
(
U1X1U
−1
1 U2X2U
−1
2
)
dU1dU2
≡
∫
U(n)
∫
U(n)
Zλ
(
U1X1U
−1
1 · U2X2U
−1
2
)
dU1dU2 =
∫
U(n)
Zλ(X1)Zλ
(
U2X2U
−1
2
)
Zλ(In)
dU2
=
Zλ(X1)
Zλ(In)
∫
U(n)
Zλ
(
U2X2U
−1
2
)
dU2.
Hyperdeterminantal Total Positivity 9
Since sλ, and hence Zλ, is unitarily invariant then Zλ
(
U2X2U
−1
2
)
= Zλ(X2) for all U2 ∈ U(n).
Also since the Haar measure dU2 is normalized then we obtain∫
U(n)
∫
U(n)
Zλ
(
U1X1U
−1
1 U2X2U
−1
2
)
dU1dU2 =
Zλ(X1)Zλ(X2)
Zλ(In)
. (4.9)
We now state and prove the promised hyperdeterminantal generalization of (4.2).
Theorem 4.2. For k = 1, . . . , 2m, let Xk be an n × n Hermitian matrix with eigenvalues
xk,1, . . . , xk,n, and denote sλ(xk,1, . . . , xk,n) by sλ(Xk). Then∫
U(n)
· · ·
∫
U(n)
exp
(
tr
2m∏
k=1
UkXkU
−1
k
)
2m∏
k=1
dUk
= βn
∑
λ
1∏n
j=1(λj + n− j)!
· 1
(sλ(In))2m−2
2m∏
k=1
sλ(Xk). (4.10)
Proof. It follows from (4.7) that for any matrix X that is a product of Hermitian matrices,
exp(trX) =
∞∑
k=0
(trX)k
k!
=
∑
λ
1
|λ|!
Zλ(X).
Applying this formula to expand the integrand in (4.10), we obtain
exp
(
tr
2m∏
k=1
UkXkU
−1
k
)
=
∑
λ
1
|λ|!
Zλ
(
2m∏
k=1
UkXkU
−1
k
)
. (4.11)
We now apply the mean-value property (4.8) to integrate iteratively with respect to the normal-
ized Haar measures dU1, . . . ,dU2m, as in the derivation of (4.9). Then we obtain∫
U(n)
· · ·
∫
U(n)
Zλ
(
2m∏
k=1
UkXkU
−1
k
)
2m∏
k=1
dUk = Zλ(X1)
2m∏
k=2
Zλ(Xk)
Zλ(In)
, (4.12)
and it follows from (4.11) and (4.12) that∫
U(n)
· · ·
∫
U(n)
exp
(
tr
2m∏
k=1
UkXkU
−1
k
)
2m∏
k=1
dUk =
∑
λ
1
|λ|!
Zλ(X1)
2m∏
k=2
Zλ(Xk)
Zλ(In)
. (4.13)
Now using (4.6) to express each Zλ in terms of sλ, and substituting from (4.5) for ωλ, the
right-hand side of (4.13) becomes
∑
λ
ωλ
|λ|!
sλ(X1)
2m∏
k=2
sλ(Xk)
sλ(In)
=
∑
λ
sλ(In)
(n)λ
sλ(X1)
2m∏
k=2
sλ(Xk)
sλ(In)
=
∑
λ
1
(n)λ
sλ(X1) · · · sλ(X2m)
(sλ(In))2m−2
. (4.14)
Next, it is straightforward to verify that
(n)λ =
n∏
j=1
(n− j + 1)λj
=
n∏
j=1
Γ(n+ λj − j + 1)
Γ(n− j + 1)
= β−1
n
n∏
j=1
(λj + n− j)!,
and by substituting this expression for (n)λ into (4.14), we obtain (4.10). ■
10 K.W. Johnson and D.St.P. Richards
As a further generalization of Example 3.2, we have the following.
Example 4.3. LetX = {0, 1, 2, . . . }, the set of nonnegative integers. Let xk = (xk,1, . . . , xk,n) ∈
Cn, k = 1, . . . , 2m and, for i ∈ X, define ϕk,r(i) = xik,r, 1 ≤ k ≤ 2m, 1 ≤ r ≤ n. For nonnegative
integers p and q, let a1, . . . , ap, b1, . . . , bq ∈ C such that −bt + j − 1 is not a nonnegative integer
for all t = 1, . . . , q and j = 1, . . . , n.
Define the discrete measure, µ({i}) = (a1)i···(ap)i
(b1)i···(bq)i
1
i! , i ∈ X, and also define the multidimen-
sional array A(r1, . . . , r2m) according to (3.1) so that
A(r1, . . . , r2m) =
∞∑
i=0
(a1)i · · · (ap)i
(b1)i · · · (bq)i
(x1,r1 · · ·x2m,r2m)
i
i!
≡ pFq(a1, . . . , ap; b1, . . . , bq;x1,r1 · · ·x2m,r2m), (4.15)
where
pFq(a1, . . . , ap; b1, . . . , bq;x) =
∞∑
i=0
(a1)i · · · (ap)i
(b1)i · · · (bq)i
xi
i!
, (4.16)
x ∈ C, is the standard notation for the classical generalized hypergeometric series [1].
The convergence properties of the series (4.15) follow from standard criteria for convergence
of the classical generalized hypergeometric series [1, Theorem 2.1.1]: For p ≤ q, the series (4.15)
converges for all x1,r1 , . . . , x2m,r2m ∈ C; if p = q + 1, then the series converges whenever
|x1,r1 · · ·x2m,r2m | < 1; and if p > q + 1, then the series diverges for all x1,r1 · · ·x2m,r2m ̸= 0
unless it is a terminating series.
Next, we apply (3.8), viz., det
(
xisk,r
)
1≤r,s≤n
= V (xk)sλ(xk), to obtain
Det(pFq(a1, . . . , ap; b1, . . . , bq;x1,r1 · · ·x2m,r2m))∏2m
k=1 V (xk)
=
∑
λ
n∏
j=1
(a1)λj+n−j · · · (ap)λj+n−j
(b1)λj+n−j · · · (bq)λj+n−j
· 1
(λj + n− j)!
·
2m∏
k=1
sλ(xk). (4.17)
It was assumed earlier that −bt + j − 1 is not a nonnegative integer for all t = 1, . . . , q and
all j = 1, . . . , n; that assumption is necessary to ensure that (bt)λj+n−j ̸= 0 for all t = 1, . . . , q
and j = 1, . . . , n, in which case the expansion in (4.17) is well defined.
By (4.3) and (4.4),
n∏
j=1
(a)λj+n−j =
n∏
j=1
Γ(a+ λj + n− j)
Γ(a)
=
n∏
j=1
(a+ n− j)λj
Γ(a+ n− j)
Γ(a)
= (a+ n− 1)λ
n∏
j=1
(a)n−j ,
so we obtain
Det(pFq(a1, . . . , ap; b1, . . . , bq;x1,r1 · · ·x2m,r2m))∏2m
k=1 V (xk)
=
n∏
j=1
(a1)n−j · · · (ap)n−j
(b1)n−j · · · (bq)n−j
×
∑
λ
(a1 + n− 1)λ · · · (ap + n− 1)λ
(b1 + n− 1)λ · · · (bq + n− 1)λ
·
n∏
j=1
1
(λj + n− j)!
·
2m∏
k=1
sλ(xk). (4.18)
Hyperdeterminantal Total Positivity 11
For (p, q) = (0, 0), since 0F0(x) = exp(x), x ∈ C (see (4.15)), then (4.18) reduces to (3.9) and
we recover Example 3.2.
For (p, q) = (1, 0), we apply the classical negative-binomial theorem, 1F0(a;x) = (1 − x)−a,
|x| < 1, and then we obtain from (4.18) the formula
Det((1− x1,r1 · · ·x2m,r2m)
−a)∏2m
k=1 V (xk)
=
n∏
j=1
(a)n−j ·
∑
λ
(a+ n− 1)λ
n∏
j=1
1
(λj + n− j)!
·
2m∏
k=1
sλ(xk).
For the case in which a = 1, this identity reduces to [30, Corollary 3.2].
We now extend Theorem 4.2 to arbitrary generalized hypergeometric functions of Hermitian
matrix argument [14, 15]. Let a1, . . . , ap and b1, . . . , bq be complex numbers such that −bt+j−1
is not a nonnegative integer for all t = 1, . . . , q and j = 1, . . . , n. The generalized hypergeometric
function of an n× n Hermitian matrix argument X is defined by the zonal polynomial series
pFq(a1, . . . , ap; b1, . . . , bq;X) =
∞∑
l=0
1
l!
∑
|λ|=l
(a1)λ · · · (ap)λ
(b1)λ · · · (bq)λ
Zλ(X), (4.19)
where the inner sum is over all partitions λ of all nonnegative integers such that λ is of
length ℓ(λ) ≤ n and weight |λ| = l.
For the case in which n = 1, the series (4.19) reduces to the classical, scalar-argument case
in (4.16). Although the notation pFq is used in both the scalar and matrix argument cases, its
usage will always be clear from the context.
The convergence properties of the series (4.19), established in [14, Section 6] in terms of ∥X∥,
the spectral norm of X, are as follows: For p ≤ q, the series (4.19) converges for all ∥X∥ < ∞;
if p = q + 1, then the series converges whenever ∥X∥ < 1; and if p > q + 1, then the series
diverges unless it is terminating.
If (p, q) = (0, 0), then by (4.7)
0F0(X) =
∞∑
l=0
1
l!
∑
|λ|=l
Zλ(X) =
∞∑
l=0
1
l!
(trX)l = exp(trX). (4.20)
For (p, q) = (1, 0), a ∈ C, and ∥X∥ < 1, it is well known (see, e.g., [14, 15]) that
1F0(a;X) = det(In −X)−a, (4.21)
where In denotes the n× n identity matrix.
The following result extends Theorem 4.2 to any generalized hypergeometric functions of
matrix argument. The proof of this result is obtained by expanding the matrix argument pFq
function and applying the same integration and simplification procedures used in (4.11)–(4.13)
in the course of proving Theorem 4.2, together with the iterative integration argument used to
derive (4.9).
Theorem 4.4. For k = 1, . . . , 2m, let Xk be an n × n Hermitian matrix with eigenvalues
xk,1, . . . , xk,n. Then∫
U(n)
· · ·
∫
U(n)
pFq
(
a1, . . . , ap; b1, . . . , bq;
2m∏
k=1
UkXkU
−1
k
)
2m∏
k=1
dUk
= βn
∑
λ
(a1)λ · · · (ap)λ
(b1)λ · · · (bq)λ
1∏n
j=1(λj + n− j)!
1
(sλ(In))2m−2
2m∏
k=1
sλ(Xk). (4.22)
12 K.W. Johnson and D.St.P. Richards
On setting (p, q) = (0, 0) in (4.22) and applying (4.20), we recover Theorem 4.2. Further,
if (p, q) = (1, 0), then by applying (4.21) we obtain
∫
U(n)
· · ·
∫
U(n)
det
(
In −
2m∏
k=1
UkXkU
−1
k
)−a 2m∏
k=1
dUk
= βn
∑
λ
(a)λ
1∏n
j=1(λj + n− j)!
1
(sλ(In))2m−2
2m∏
k=1
sλ(Xk),
with convergence for max{∥X1∥, . . . ∥X2m∥} < 1.
5 Hyperdeterminantal total positivity
In this section, we define the concept hyperdeterminantal total positivity, construct several
examples of such kernels, and generalize the classical basic composition formula. Further, we
apply the hyperdeterminantal Binet–Cauchy formula and the generalized basic composition
formula to show how examples of HTP kernels can be constructed.
5.1 Definition of hyperdeterminantal total positivity
We now define the concept of hyperdeterminantal total positivity of general order d for a ker-
nel K : R2m → [0,∞). As in Section 3, we use the notation
Cn = {(x1, . . . , xn) ∈ Rn | x1 > · · · > xn}
for the fundamental Weyl chamber in Rn.
Definition 5.1. A kernel K : R2m → [0,∞) is hyperdeterminantal totally positive of order d
(HTPd) if, for any collection of vectors xk = (xk,1, . . . , xk,n) ∈ Cn, 1 ≤ k ≤ 2m, we have
Det(K(x1,r1 , x2,r2 , . . . , x2m,r2m))1≤r1,...,r2m≤n ≥ 0 (5.1)
for all n = 1, . . . , d. If the hyperdeterminants (5.1) are nonnegative for all n ∈ N, then we say
that K is hyperdeterminantal totally positive of order ∞ (HTP∞).
If the hyperdeterminants (5.1) are positive for all n = 1, . . . , d, then we say that K is hyper-
determinantal strictly totally positive of order d (HSTPd). Similarly, if strict inequality holds
for all n ∈ N, then we say that K is hyperdeterminantal strictly totally positive of order infinity
(HSTP∞).
We remark that Definition 5.1 was given in [20] and, independently, in [38]. Intuitively,
a kernel K : R2m → [0,∞) is HTPd if the hyperdeterminant of every sub-array of the array
(K(x1,r1 , x2,r2 , . . . , x2m,r2m))1≤r1,...,r2m≤d (5.2)
is nonnegative for all vectors xk = (xk,1, . . . , xk,n) ∈ Cd, k = 1, . . . , 2m. Note that the concept
of a minor of an array was treated in [3] and, when expressed in that terminology, a ker-
nel K : R2m → [0,∞) is HTPd if all minors of the array (5.2) are nonnegative for all vec-
tors x1, . . . ,x2m ∈ Cd.
Hyperdeterminantal Total Positivity 13
5.2 Examples of HTP kernels
Example 5.2. The case m = 1: By the definition in (2.1) of the hyperdeterminant, a ker-
nel K : R2 → [0,∞) is HTPd if, for all n = 1, . . . , d, the determinant
det(K(x1,r1 , x2,r2))1≤r1,r2≤n =
∣∣∣∣∣∣∣∣∣
K(x1,1, x2,1) K(x1,1, x2,2) · · · K(x1,1, x2,n)
K(x1,2, x2,1) K(x1,2, x2,2) · · · K(x1,2, x2,n)
...
...
...
K(x1,n, x2,1) K(x1,n, x2,2) · · · K(x1,n, x2,n)
∣∣∣∣∣∣∣∣∣
is nonnegative whenever x1,1 > x1,2 > · · · > x1,d and x2,1 > x2,2 > · · · > x2,d. This condition is
well known to be the classical definition of total positivity of order d.
For m ≥ 2, the concept of HTPd appears to be new even for d = 2, and the following example
illustrates the difference between HTP and the concept of multivariate total positivity developed
in [23].
Example 5.3. The case d = m = 2: For vectors xk = (xk,1, xk,2) ∈ C2, k = 1, . . . , 4, we
apply (2.2) to obtain
Det(K(x1,r1 , . . . , x4,r4))1≤r1,...,r4≤2
=
∣∣∣∣K(x1,1, x2,1, x3,1, x4,1) K(x1,1, x2,1, x3,1, x4,2)
K(x1,2, x2,2, x3,2, x4,1) K(x1,2, x2,2, x3,2, x4,2)
∣∣∣∣
−
∣∣∣∣K(x1,1, x2,2, x3,1, x4,1) K(x1,1, x2,2, x3,1, x4,2)
K(x1,2, x2,1, x3,2, x4,1) K(x1,2, x2,1, x3,2, x4,2)
∣∣∣∣
−
∣∣∣∣K(x1,2, x2,1, x3,1, x4,1) K(x1,2, x2,1, x3,1, x4,2)
K(x1,1, x2,2, x3,2, x4,1) K(x1,1, x2,2, x3,2, x4,2)
∣∣∣∣
+
∣∣∣∣K(x1,2, x2,2, x3,1, x4,1) K(x1,2, x2,2, x3,1, x4,2)
K(x1,1, x2,1, x3,2, x4,1) K(x1,1, x2,1, x3,2, x4,2)
∣∣∣∣ . (5.3)
Define the vectors
u1 = (x1,1, x2,2), u2 = (x1,2, x2,1), v1 = (x3,1, x4,2), v2 = (x3,2, x4,1).
Since each xk ∈ C2, k = 1, . . . , 4, then
u1 ∨ u2 = (x1,1, x2,1), u1 ∧ u2 = (x1,2, x2,2),
v1 ∨ v2 = (x3,1, x4,1), v1 ∧ v2 = (x3,2, x4,2).
Therefore, (5.3) becomes
Det(K(x1,r1 , . . . , x4,r4))1≤r1,...,r4≤2
=
∣∣∣∣K(u1 ∨ u2,v1 ∨ v2) K(u1 ∨ u2,v1)
K(u1 ∧ u2,v2) K(u1 ∧ u2,v1 ∧ v2)
∣∣∣∣
−
∣∣∣∣K(u1,v1 ∨ v2) K(u1,v1)
K(u2,v2) K(u2,v1 ∧ v2)
∣∣∣∣
−
∣∣∣∣K(u2,v1 ∨ v2) K(u2,v1)
K(u1,v2) K(u1,v1 ∧ v2)
∣∣∣∣
+
∣∣∣∣K(u1 ∧ u2,v1 ∨ v2) K(u1 ∧ u2,v1)
K(u1 ∨ u2,v2) K(u1 ∨ u2,v1 ∧ v2)
∣∣∣∣ . (5.4)
14 K.W. Johnson and D.St.P. Richards
Let w1 = (u1 ∨ u2,v1), w2 = (u1 ∧ u2,v2). Then, w1 ∨w2 = (u1 ∨ u2,v1 ∨ v2), w1 ∧w2 =
(u1 ∧ u2,v1 ∧ v2), and the first determinant on the right-hand side of (5.4) equals∣∣∣∣K(w1 ∨w2) K(w1)
K(w2) K(w1 ∧w2)
∣∣∣∣ . (5.5)
Therefore, if K is multivariate TP2 in the sense defined in [23] then it follows that the deter-
minant (5.5) is nonnegative. However, for arbitrary multivariate TP2 kernels, it appears to be
difficult to derive general criteria that imply the nonnegativity of the alternating sum of all four
determinants in (5.4).
Turning to the construction of HTPd kernels, we remarked in the introduction that the funda-
mental example of a classical STP∞ kernel is the function K(x1, x2) = exp(x1x2), (x1, x2) ∈ R2.
We now generalize this example to the setting of hyperdeterminantal total positivity.
Define a slice of the array A = (A(r1, . . . , r2m))1≤r1,...,r2m≤n in the jth direction to be the
subset of all indices (r1, . . . , r2m) where the jth index, rj , is fixed. Gelfand, et al. [12, Corol-
lary 1.5 (b)] used the action of certain general linear groups to show that Det(A) is a homogeneous
polynomial in the entries of each slice; this homogeneity property can also be deduced directly,
albeit with more work, from the expansions (2.2) and (2.3).
Proposition 5.4. The following kernels are HSTP∞:
(i) K1(x1, . . . , x2m) = exp(x1 · · ·x2m), x1, . . . , x2m ∈ R.
(ii) K2(x1, x2, . . . , x2m) = xx2x3···x2m
1 , x1, x2, . . . , x2m > 0.
Proof. (i) Let xk = (xk,1, . . . , xk,n), k = 1, . . . , 2m, and suppose that each xk ∈ [0,∞)n, the
nonnegative orthant in Rn. By (3.9),
Det(exp(x1,r1x2,r2 · · ·x2m,r2m)) =
(
2m∏
k=1
V (xk)
)
·
∑
λ
n∏
j=1
1
(λj + n− j)!
·
2m∏
k=1
sλ(xk), (5.6)
where the sum is over all partitions λ = (λ1, . . . , λn) of length ℓ(λ) ≤ n. It is a well-known
result [15, 29] that sλ(xk) ≥ 0 for all xk ∈ [0,∞)n, and therefore each summand on the right-
hand side of (5.6) is nonnegative. Moreover, the sum is positive since s(0)(xk) ≡ 1, where (0)
denotes the zero partition.
Suppose also that each xk ∈ Cn. Then, by (3.6), the Vandermonde polynomials V (xk) on
the right-hand side of (5.6) are positive. Therefore, the right-hand side of (5.6) is positive
for xk ∈ Cn ∩ [0,∞)n, k = 1, . . . , 2m, so we obtain
Det(exp(x1,r1x2,r2 · · ·x2m,r2m)) > 0.
This proves that the kernel K1 is SHTPn, and since n was chosen arbitrarily then it follows
that K1 is HSTP∞.
Now consider the case in which xk,n < 0 for some k. For each such k, we multiply the
corresponding slice of the array, viz.,
{(exp(x1,r1x2,r2 · · ·x2m,r2m)) | 1 ≤ i1, . . . , i2m ≤ n, ik = n}
by exponential factors as follows: Form the vectors xk,nxi = (xk,nxk,1, . . . , xk,nxin), 1 ≤ i ≤ n;
next, multiply the corresponding slice by
exp(−x1,i1x2,i2 · · ·xk−1,ik−1
xk,nxk+1,ik+1
· · ·x2m,i2m).
Hyperdeterminantal Total Positivity 15
thereby changing the generic entries of that slice to
exp(−x1,i1x2,i2 · · ·xk−1,ik−1
(xk,ik − xk,n)xk+1,ik+1
· · ·x2m,i2m).
On carrying out these multiplications for each k such that xk,n < 0, the outcome is that we have
created a new multidimensional array, (exp(y1,r1y2,r2 · · · y2m,r2m)), where
yk,rk =
{
xk,rk if xk,n ≥ 0,
xk,rk − xk,n if xk,n < 0.
Noting that yk,rk ≥ 0 for all k = 1, . . . , 2m, it then follows from the preceding argument
that Det(exp(y1,r1y2,r2 · · · y2m,r2m)) > 0.
As noted earlier, the hyperdeterminant is a homogeneous polynomial in the entries of each
slice. Hence Det(exp(y1,r1y2,r2 · · · y2m,r2m)) and Det(exp(x1,r1x2,r2 · · ·x2m,r2m)) differ only by
powers of the multiplying factors. Since those factors are all exponential terms then the hy-
perdeterminants Det(exp(y1,r1y2,r2 · · · y2m,r2m)) and Det
(
(x1,r1x2,r2 · · ·x2m,r2m)) differ only by
positive factors, and therefore Det(exp(x1,r1x2,r2 · · ·x2m,r2m)) > 0.
(ii) Suppose that x1, x2, . . . , x2m > 0. We now make the change-of-variables x1 → log x1 in
the kernel in (i). Then the resulting kernel is K2(x1, x2, . . . , x2m) = xx2x3···x2m
1 , and it follows
from the HDTP∞ property of K1 and strictly increasing nature of the mapping x1 → log x1
that K2 also is HDTP∞. ■
Remark 5.5. For the case in which some of the xj are integer variables, a kernel similar to K1
was studied in [38, 39].
Also, in Proposition 5.4, if x1, x2, . . . , x2m > 0 and we make the changes-of-variables xj →
exp(xj), j = 1, . . . , 2m, then the resulting kernel is K3(x1, . . . , x2m) = exp
(
ex1+···+x2m
)
, where
x1, x2, . . . , x2m ∈ R. As K3 depends on x1 + · · · + x2m only, then it generalizes the Pólya
frequency function type of TP kernels. See [28, 38, 39] for kernels of this type.
Example 5.6. Consider the classical case in which m = 1. We want to show that if x1,1 >
· · · > x1,n and x2,1 > · · · > x2,n, then the determinant∣∣∣∣∣∣∣
exp(x1,1x2,1) exp(x1,1x2,2) · · · exp(x1,1x2,n)
...
...
...
exp(x1,nx2,1) exp(x1,nx2,2) · · · exp(x1,nx2,n)
∣∣∣∣∣∣∣
is positive.
If x1,n ≥ 0 and x2,n ≥ 0, then the strict positivity of the determinant follows from (3.9)
with m = 1.
For x2,n < 0 < x1,n, we multiply the jth row of the determinant by exp(−x1,jx2,n), j =
1, . . . , n. This converts the determinant to∣∣∣∣∣∣∣
exp(x1,1(x2,1 − x2,n)) exp(x1,1(x2,2 − x2,n)) · · · exp(x1,1(x2,n − x2,n))
...
...
...
exp(x1,n(x2,1 − x2,n)) exp(x1,n(x2,2 − x2,n)) · · · exp(x1,n(x2,n − x2,n))
∣∣∣∣∣∣∣ .
On the other hand, if x1,n < 0 < x2,n, then we multiply the jth column by exp(x1,nx2,j),
j = 1, . . . , n; and if x1,n < 0 and x2,n < 0, then we carry out both multiplications.
The outcome of these multiplications is to transform the original determinant to a new
determinant of the same type, with generic entries exp(y1,iy2,j) where y1,1 > y1,2 > · · · > y1,n ≥ 0
and y2,1 > y2,2 > · · · > y2,n ≥ 0, and we have seen before that the resulting determinant is
positive. Moreover, the two determinants differ in value only by positive (exponential) factors,
therefore the initial determinant is positive.
16 K.W. Johnson and D.St.P. Richards
For nonnegative integers p and q, denote by Dp,q the set of all (x1, . . . , x2m) ∈ R2m such
that the generalized hypergeometric series pFq(a1, . . . , ap; b1, . . . , bq;x1 · · ·x2m) converges. The
next result is motivated by Example 4.3, where the hyperdeterminant was constructed using this
generalized hypergeometric series, and it extends the previous example involving the exponential
function.
Proposition 5.7. Let a1, . . . , ap ≥ 0 and b1, . . . , bq > 0. The kernel K : [0,∞)2m → R such that
K(x1, . . . , x2m) = pFq(a1, . . . , ap; b1, . . . , bq;x1 · · ·x2m)
is HTP∞ on the region Dp,q ∩ [0,∞)2m.
The proof of Proposition 5.7 follows from the hyperdeterminantal summation formula (4.17)
given in Example 4.3.
Example 5.8. In Proposition 5.7, suppose that (p, q) = (1, 0) and a1 ≡ a ≥ 0. It is well
known that 1F0(a;x) = (1 − x)−a, |x| < 1, and therefore K(x1, . . . , x2m) = (1 − x1 · · ·x2m)−a,
|x1 · · ·x2m| < 1. Hence the kernelK is HTP∞ on the region {(x1, . . . , x2m) | x1 · · ·x2m < 1, x1 ≥
0, . . . , x2m ≥ 0}.
5.3 The basic composition formula
In the classical setting, examples of kernels that are hyperdeterminantal totally positive often
are constructed using the basic composition formula [21]: If K1,K2 : R → R are TPd kernels,
then so is the kernel
K(x1,1, x2,1) =
∫
R
K1(x1,1, t)K2(t, x2,1)dµ(t),
where µ is a positive Borel measure such that the integral converges absolutely. In the hyperde-
terminantal setting, the following generalization of the basic composition formula follows from
the hyperdeterminantal Binet–Cauchy formula in Proposition 3.1.
Proposition 5.9. Let (X,µ) be a totally ordered sigma-finite measure space, and let {ϕk,r |
1 ≤ k ≤ 2m, 1 ≤ r ≤ d} be a collection of real-valued functions on X. Suppose for each
k = 1, . . . , 2m that the kernel Kk(t, r) = ϕk,r(t), (t, r) ∈ X × {1, . . . , d}, is TPd. Define the
multidimensional array
A(r1, . . . , r2m) =
∫
X
2m∏
k=1
ϕk,rk(x)dµ(x),
where 1 ≤ r1, . . . , r2m ≤ d, and the integral is assumed to converge absolutely. Then the ar-
ray (A(r1, . . . , r2m))1≤r1,...,r2m≤d is HTPd.
Proof. For any n such that 1 ≤ n ≤ d, consider the hyperdeterminantal minor
Det(A(r1, . . . , r2m))1≤r1,...,r2m≤n.
By applying (3.3) in Proposition 3.1, we can represent this minor as an integral whose integrand
is nonnegative, hence the minor is nonnegative. Since n is arbitrarily chosen, then it follows
that the array (A(r1, . . . , r2m))1≤r1,...,r2m≤d is HTPd. ■
Hyperdeterminantal Total Positivity 17
As a consequence of this result, numerous HTP kernels can be constructed. For example, by
choosing ϕk,r(t) = tk+λr , where λ1, . . . , λr are the parts of a partition λ, then the resulting mi-
nor Det(A(r1, . . . , r2m))1≤r1,...,r2m≤n is a hyperdeterminantal generalization of the Vandermonde
determinant, and we obtain positivity results similar to those given in [38]. If we choose ϕk,r(t)
as a classical generalized hypergeometric series and if the measure µ is chosen suitably, then
the entries A(r1, . . . , r2m), 1 ≤ r1, . . . , r2m ≤ n, of the resulting array can be made to contain
a wide range of the classical special functions, such as the Bessel or the Gaussian hypergeo-
metric series. By choosing ϕk,r(t) in terms of the bivariate normal distributions with negative
correlations [23], the entries of the resulting array can be made to include numerous multivariate
probability distributions.
6 Concluding remarks
The results presented in this article raise many open problems in a variety of areas. For the pur-
poses of making applications in statistics and probability, as in [23], we are especially interested
in hyperdeterminantal generalizations of the FKG inequality. Here, even the most basic ques-
tions are still open, e.g., the determination of conditions under which the familiar multivariate
probability distributions, such as the multivariate normal distributions, are HTP of any order.
A second problem arises from the definition of the hyperdeterminant. Suppose that the sym-
metric group Sn in the sums that define Det(K(x1,r1 , x2,r2 , . . . , x2m,r2m)) in (5.1) were to be
replaced by an arbitrary finite reflection group. Following the approach in [16], corresponding
generalizations of the Binet–Cauchy formula can be obtained and examples of hyperdetermi-
nantal totally positive kernels can be constructed. It would be a fundamental achievement to
derive analogs of the FKG inequality in this setting.
Acknowledgements
The results in this article were first presented in [20] at the Second CREST-SBM International
Conference, “Harmony of Gröbner Bases and the Modern Industrial Society”, held June 28 –
July 2, 2010, in Osaka, Japan (see [18]). We express our gratitude to the organizers of the
meeting for the opportunity to have presented our results there.
We are also grateful to SIGMA’s referees for their meticulous reading of the article and very
helpful comments.
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1 Introduction
2 Hyperdeterminants
3 A Binet–Cauchy theorem for hyperdeterminants
4 A hyperdeterminantal extension of an integral of Harish-Chandra
5 Hyperdeterminantal total positivity
5.1 Definition of hyperdeterminantal total positivity
5.2 Examples of HTP kernels
5.3 The basic composition formula
6 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-213521 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T16:46:49Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Johnson, Kenneth W. Richards, Donald St. P. 2026-02-18T11:24:01Z 2025 Hyperdeterminantal Total Positivity. Kenneth W. Johnson and Donald St. P. Richards. SIGMA 21 (2025), 055, 18 pages 1815-0659 2020 Mathematics Subject Classification: 33C20; 05E05; 15A15; 15A72; 33C80 arXiv:2412.03000 https://nasplib.isofts.kiev.ua/handle/123456789/213521 https://doi.org/10.3842/SIGMA.2025.055 For a given positive integer , the concept of hyperdeterminantal total positivity is defined for a kernel : ℝ²ᵐ → ℝ, thereby generalizing the classical concept of total positivity. Extending the fundamental example, (, ) = exp(), , ∈ ℝ, of a classical totally positive kernel, the hyperdeterminantal total positivity property of the kernel (₁, …, ₂ₘ) = exp(₁⋯₂ₘ), ₁,…,₂ₘ ∈ ℝ is established. By applying Matsumoto's hyperdeterminantal Binet-Cauchy formula, we derive a generalization of Karlin's basic composition formula. We use the generalized composition formula to construct several examples of hyperdeterminantal totally positive kernels. Further generalizations of hyperdeterminantal total positivity via the theory of finite reflection groups are described, and some open problems are posed. The results in this article were first presented in [20] at the Second CREST-SBM International Conference, “Harmony of Gröbner Bases and the Modern Industrial Society”, held June 28 July 2, 2010, in Osaka, Japan (see [18]). We express our gratitude to the organizers of the meeting for the opportunity to present our results there. We are also grateful to SIGMA’s referees for their meticulous reading of the article and very helpful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hyperdeterminantal Total Positivity Article published earlier |
| spellingShingle | Hyperdeterminantal Total Positivity Johnson, Kenneth W. Richards, Donald St. P. |
| title | Hyperdeterminantal Total Positivity |
| title_full | Hyperdeterminantal Total Positivity |
| title_fullStr | Hyperdeterminantal Total Positivity |
| title_full_unstemmed | Hyperdeterminantal Total Positivity |
| title_short | Hyperdeterminantal Total Positivity |
| title_sort | hyperdeterminantal total positivity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213521 |
| work_keys_str_mv | AT johnsonkennethw hyperdeterminantaltotalpositivity AT richardsdonaldstp hyperdeterminantaltotalpositivity |