From Cauchy's determinant formula to bosonic and fermionic immanant identities

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European Journal of Combinatorics

journal homepage:www.elsevier.com/locate/ejc

From Cauchy’s determinant formula to bosonic and fermionic immanant identities

Apoorva Khare

^a,b

, Siddhartha Sahi

^c

aDepartment of Mathematics, Indian Institute of Science, Bangalore, 560012, India

bAnalysis and Probability Research Group, Bangalore, 560012, India

cDepartment of Mathematics, Rutgers University, Piscataway 08854, USA

a r t i c l e i n f o

Article history:

Received 1 March 2022 Accepted 19 December 2022 Available online xxxx

a b s t r a c t

Cauchy’s determinant formula (1841) involving det((1−u_ivj)⁻¹) is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius (1882) involving a sum of two geometric series in uivj. This theme also resurfaced in a matrix analysis setting in a paper by Horn (1969) – where the computations are attributed to Loewner – and in recent works by Belton et al.

(2016) and Khare and Tao (2021). These formulas were recently unified and extended in Khare (2022) to arbitrary power series, with commuting/bosonic variablesu_i, vj.

In this note we formulate analogous permanent identities, and in fact, explain how all of these results are a special case of a more general identity, for any character – in fact, any complex class function – of any finite group that acts on the bosonic variablesu_iand on thevj via signed permutations. (We explain why larger linear groups do not work, via a – perhaps novel – ‘‘symmetric function’’ characterization of signed permutation matrices that holds over any integral domain.) We then provide fermionic analogues of these formulas, as well as of the closely related Cauchy product identities.

©2022 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Post-1960 results: Entrywise positivity preservers and schur polynomials

The goal of this note is to extend some classical and modern symmetric function determinantal identities to other characters of the symmetric group (and its subgroups), and then to formulate and

E-mail addresses: khare@iisc.ac.in(A. Khare),sahi@math.rutgers.edu(S. Sahi).

https://doi.org/10.1016/j.ejc.2022.103683

0195-6698/©2022 Elsevier Ltd. All rights reserved.

show fermionic counterparts of these. The origins of this work lie in classical identities by Cauchy and Frobenius, but also in a computation – see Theorem 1.1– that originally appears in a letter by Charles Loewner to Josephine Mitchell on October 24, 1967 (as observed by the first-named author in the Stanford Library archives). Subsequently, this computation, and the broader result on

‘‘entrywise functions’’, appeared in print in the thesis of Loewner’s Ph.D. student, Roger Horn — see also the proof of [6, Theorem 1.2], which Horn attributes to Loewner.

In his letter, Loewner explained that he was interested in understanding functions actingentry- wiseon positive semidefinite matrices (i.e., real symmetric matrices with non-negative eigenvalues) of a fixed size, and preserving positivity. Previously, results by Schur, Schoenberg, and Rudin had classified the dimension-free preservers, i.e., the entrywise maps preserving positivity inall dimensions [19,21,22]. In contrast, in a fixed dimensiond, such a classification remains open to date, even ford

=

3; moreover, Loewner’s 1967 result is still state-of-the-art, in that it is (essentially) the only known necessary condition for a general entrywise function preserving positivity in a fixed dimension. We refer the reader to e.g. [10] for more details.

The present work begins by isolating from Loewner’s positivity/analysis result, the following algebraic calculation. Fix an integern⩾ 2; given a matrixA

=

(a_ij), here and belowf

[

A

]

denotes the matrix with (i

,

^j)-entryf(aij).

Theorem 1.1(Loewner).Suppose f

:

_R

→

_Ris a smooth function, n⩾2, and letu

=

(u₁

, . . . ,

^un)^T

∈

Rⁿ. Define the determinant function

∆

:

_R

→

_R

,

^t

↦→

det(f(tu_iu_j))ⁿ_i,j=1

=

detf

[

tuu^T

] .

Then∆⁽⁰⁾

= · · · =

_∆

(

ⁿ2

)

⁻¹(0)

=

0, and the next derivative is

∆

(

ⁿ2

)

₍₀₎

=

( (_n

2

) 1

,

²

, . . . ,

ⁿ

−

1

)

∏

i<j

(u_j

−

u_i)²

·

f(0)f^′(0)

· · ·

f⁽ⁿ⁻¹⁾(0)

.

^(1.1) In particular, iff(t) is a convergent power series∑

n⩾0f_ntⁿ, then within a suitable radius of convergence,

∆^(t)

=

t

(

ⁿ2

)

∏

i<^j

(u_j

−

u_i)²

·

f₀f₁

· · ·

f_n−₁

+

higher order terms

.

The first term on the right-hand side of Eq. (1.1)is a multinomial coefficient, and the reader will recognize the next product as the square of a Vandermonde determinant for the matrix with entriesuⁿ_i^−j

,

¹⩽i

,

^j⩽n. What the reader may find harder to recognize is that Eq.(1.1)contains a

‘‘hidden’’ Schur polynomial (these are defined presently) in the variablesu_i: the simplest of them all, s_(0,...,0)(u)

=

1. In particular, if one goes even one derivative beyond Loewner’s stopping point, one immediately uncovers other, nontrivial Schur polynomials. This is stated precisely inTheorem 1.2.

The presence of the lurking (simplest) Schur polynomial in(1.1)was suspected owing to very recent sequels to Loewner’s matrix positivity result. First with Belton–Guillot–Putinar [2] and then with Tao [11], the first-named author found (the first) examples of polynomial maps with at least one negative coefficient, which preserve positivity in a fixed dimension when applied entrywise.

These papers uncovered novel connections between polynomials that entrywise preserve positivity and Schur polynomials, and in particular, obtained expansions for detf

[

tuv^T

]

in terms of Schur polynomials, for all polynomialsf(t). This suggested revisiting the general case due to Loewner (in slightly greater generality, as above: for detf

[

tuv^T

]

).

1.2. Pre-1900 results: Cauchy and Frobenius

We now go back in history and remind the reader of the first such determinantal identities involving Schur polynomials. Recall the well-known Cauchy determinant identity [3], [17, Chapter I.4, Example 6]: ifBis then

×

nmatrix with entries (1

−

u_i

v

j)⁻¹

=

∑

M⩾0(u_i

v

j)^Mfor variablesu_i

, v

j

with 1⩽i

,

^j⩽n, then detB

=

V(u)V(v)∑

m

sm(u)s_m(v)

,

^(1.2)

2

whereV(u) for a finite tuple u

=

(u_i)_i⩾1 denotes the ‘‘Vandermonde determinant’’∏

i<j(u_j

−

u_i), and the sum runs over all partitionsmwith at mostnparts. Here, a partitionm

=

(m₁

, . . . ,

^mn) simply means a weakly decreasing sequence of nonnegative integersm₁⩾

· · ·

_⩾m_n⩾0; and we use Cauchy’s definition [9] for the Schur polynomials_m(v), namely,

s_m(

v

1

, . . . , v

n)

:=

^det(

v

_j^mi+n−i)n_i,j=₁

det(

v

_jⁿ⁻ⁱ⁾ⁿ_i,j=₁

.

(This definition differs from that in the literature, e.g. in [17].) Here and below, we restrict ton arguments

v

j, to go with thenexponentsm_i.

See also [14, Section 5] and the references therein, as well as [7,8,12,13,15,17] for other determinantal identities involving symmetric functions.

As discussed in Section 1.1, in this paper we focus on the specific form of the determinant in (1.2), i.e. where one applies to all u_i

v

j some power series (Eq. (1.2) considers the case of f(x)

=

1

/

⁽¹

−

x)

=

∑

M⩾0x^M), and then computes the determinant. For instance, iff(x) has fewer thannmonomials thenf

[

uv^T

]

is a sum of fewer thannrank-one matrices, hence is singular. (For more general polynomials – as mentioned above – the formula was worked out in [11].) Another such formula was shown by Frobenius [4], in fact in greater generality.¹ The formula appears in Rosengren–Schlosser [18, Corollary 4.7] as well, as a consequence of their Theorem 4.4; and it implies a more general determinantal identity than(1.2), with (1

−

cx)

/

⁽¹

−

x) replacing 1

/

⁽¹

−

x) and the sum again running over all partitions with at mostnparts:

det

(1

−

cu_i

v

j

1

−

u_i

v

j

)n i,^j=₁

(1.3)

=

V(u)V(v)(1

−

c)ⁿ⁻¹ (

∑

m:_mn=₀

s_m(u)s_m(v)

+

(1

−

c) ∑

m:_mn>0

s_m(u)s_m(v) )

.

1.3. The present work

Given the many precursors listed above, it is natural to seek a more general identity, i.e. the expansion of detf

[

uv^T

]

, wheref

[

uv^T

]

is the entrywise application of an arbitrary (formal) power series f to the rank-one matrix uv^T

=

(u_i

v

j)ⁿ_i,j=1. This question was recently answered by the first-named author – including additional special cases – again in the context of matrix positivity preservers.

Theorem 1.2(Khare, [10]).Fix a unital commutative ring R and let t be an indeterminate. Let f(t)

:=

∑

M⩾0f_Mt^M

∈

R

[[

t

]]

be an arbitrary formal power series. Given vectorsu

,

^v

∈

Rⁿfor some n⩾1, we have:

detf

[

tuv^T

] =

V(u)V(v)∑

M⩾0

t^M+

(

ⁿ2

)

∑

m=_(m1,...,mn)⊢_M

s_m(u)s_m(v)

n

∏

i=₁

f_mi+_n−_i

,

^(1.4) wherem

⊢

M means thatmis a partition whose components sum to M.

The goal of this short note is to show that these identities hold more generally — not just for determinants, but also e.g. for permanents, where

perm(A_n×_n)

:=

∑

σ∈_Sn

a_1σ(1)

· · ·

a_nσ(n)

.

Thus we show below:

1 Here one uses theta functions and obtains elliptic Frobenius–Stickelberger–Cauchy determinant (type) identities; see also [1,5].

3

Theorem 1.3. With notation as inTheorem1.2(and over any commutative unital ring R), we have:

n

!

permf

[

tuv^T

] =

∑

m∈Zⁿ_⩾0

t^m1+···+mnperm(u^◦m) perm(v^◦m)

n

∏

i=1

f_mi

,

wherev^◦m

:=

(

v

_j^mi)(and similarly foru^◦m), andm⩾0is interpreted coordinatewise.

We show this result as well as Theorem 1.2 by a common proof. In fact we go beyond permanents: we provide such an identity for an arbitrary character of an arbitrary subgroupGof S_n— and even of the hyperoctahedral group:G⩽S₂

≀

S_n. Thus, our proof differs from the approach in [10], and proceeds via group representation theory. We then produce a fermionic analogue of the bosonic immanant ‘‘master identity’’, in which the variablesu_i anti-commute, as do the

v

j. For quick reference, these identities are summarized in the following table (seeTable 1).

Table 1

The first three rows provide formulas for an arbitrary formal power series applied entrywise to the matrixtuv=(tuivj)ⁿ_i,j=1. The fourth row computes the product of (1−uivj)⁻¹or of (1+uivj).

Two of these formulas can be found in earlier literature, see [10,17].

Even (bosonic) variables Odd (fermionic) variables

Determinant (forSn) (2.7)(see [10]) (3.3)

Permanent (forSn) (2.8) (3.4)

Arbitrary immanants for subgroups ofS2≀Sn

(2.6) (3.2)

(Bi)Product identities (3.5)(see e.g. [17]) (3.6)

2. Immanant identities for bosonic variables

2.1. Establishing the setting

In order to state and prove our main results for arbitrary immanants of complex characters – and more generally, for their linear combinations (i.e., class functions) – we first establish the setting in which our results hold.

2.1.1. Ground ring — contains character values

The first step is to explain the degree of freedom in choosing the ground ring. Fix an integern⩾1 and a unital commutative subringR. Recall that given a complex character

χ

of the permutation groupS_n, theimmanant of a square matrixA_n×_n– as defined by Littlewood and Richardson [16] – is

Imm_χ(A)

:=

∑

σ∈_Sn

χ

⁽

σ

⁾

n

∏

i=₁

a_i,σ(i)

.

For this to act onf(tu_i

v

j) withf

∈

R

[[

t

]]

, we needRto ‘‘contain’’ the character values of

χ

^{. This is} made precise as follows:

Definition 2.1. Given a finite group G and a complex character – or class function –

ψ :=

∑

χ∈ˆG_Ca_χ

χ

(where the sum runs over the irreducible complex characters ofG), define thering of character values R_ψ

⊂

_Cto be the unital subring generated by all character values that occur in

ψ

^:

R_ψ

:=

_Z

[{ χ

^(g)

:

g

∈

G

,

^aχ

̸=

0

}] ⊂

_C

.

^(2.1)

In this paper we will work over arbitrary commutativeR_ψ-algebrasR. For instance, if

ψ = χ

is the determinant or permanent, thenR_ψ

=

_Z, and so the determinant and permanent identities will hold over every commutativeZ-algebra – i.e. unital commutative ring –R, for all power series f

∈

R

[[

t

]]

. (Hence the theorems above are stated over allR.)

4

2.1.2. The Segre subalgebra — contains the immanant

The other point to establish is the algebra in which our immanant identities are to hold. For this, we begin by reminding the reader that we will work with arbitrary subgroups of the group of signed permutationsS₂

≀

S_n. (To understand this choice of groupS₂

≀

S_nand not a larger one, see Section2.3.) Now suppose u₁

, . . . ,

^un

, v

1

, . . . , v

n are commuting variables. Recall thatS₂

≀

S_n acts on the tensor algebraR

⟨

u₁

, . . . ,

^un

⟩

via signed permutations on the span of theu_i, and similarly on R

⟨ v

1

, . . . , v

n

⟩

. Explicitly, write every elementg

∈

S₂

≀

S_nas:g

=

D_g

σ

g, whereD_g

=

D⁻_g¹is a diagonal matrix with (i

,

i) diagonal entry (D_g)_i

∈ {±

1

}

, and

σ

g

∈

S_nis a permutation matrix. Now:

g

·

n

∑

i=₁

r_iu_i

:=

n

∑

i=₁

r_i

·

(D_g)_σg(i)

·

u_σg(i)

=

n

∑

j=₁

r_σ⁻1

g (j)

·

(D_g)_j

·

u_j

,

^(2.2)

and this is extended multiplicatively toR

⟨

u

⟩

. Moreover, this action preserves the two-sided ideals generated by

{

u_i

⊗

u_j

−

u_j

⊗

u_i

:

1_⩽i

,

^j⩽n

} , {

u_i

⊗

u_j

+

u_j

⊗

u_i

:

1_⩽i

<

^j⩽n

} ⊔ {

u_i

⊗

u_i

:

1_⩽i_⩽n

} .

Hence denoting the freeR-moduleU

:=

∑n

i=₁Ru_i, the action ofS₂

≀

S_nonR

⟨

u

⟩

descends to the quotient symmetric algebra Sym^•_R(U) and quotient alternating algebra

∧

^•_R(U), via:

g(u^m)

=

n

∏

i=₁

((D_g)_σg(i)

·

u_σg(i))^mi

,

g(u_i1

∧ · · · ∧

u_id)

=

d

∏

j=₁

(D_g)_σg(ij)

·

(u_σg(i1)

∧ · · · ∧

u_σg(id))

,

(2.3)

for all non-negative integer tuplesm

=

(m₁

, . . . ,

^mn) and all integers 1 ⩽ i₁

< · · · <

ⁱd ⩽ n.

Here and below, we define and useu^m

:=

∏n

i=₁u^m_iⁱ. Notice that this action preserves each graded component

Sym^d_R(u)

:=

Sym^d_R(U)

, ∧

^d_R(u)

:= ∧

^d_R(U)

,

^d⩾0

.

^(2.4) The above holds verbatim for the

v

j, i.e. for the freeRmoduleV

:=

∑n

j=₁R

v

j. This action carries over to the tensor algebraR

⟨

v

⟩

and its symmetric/alternating quotients.

Now fix a finite subgroupG⩽S₂

≀

S_n, an irreducible complex character

χ

^ofG, and a commutative R_χ-algebraR. Recall that one has the ‘‘minimal’’ pseudo-idempotent in the group algebra

E_χ

:= χ

⁽¹⁾∑

g∈_G

χ

^(g)g−1

∈

R_χG

(see(2.1)), where ‘‘pseudo-idempotent’’ simply means thatE_χ²

= |

G

|

E_χ inR_χ (and hence inR_ψ for any class function

ψ

^with

[ ψ : χ ] ̸=

0). Below, we will act by E_χ on both theu_i and on the

v

j, and so to keep track of which variables are acted upon, we denote byE_χ^u the pseudo-idempotent acting on R

⟨

u

⟩

, and hence on each u^m and each

∧

^k_j=1u_ij via (2.3). Similarly,E_χ^v will denote the pseudo-idempotent acting onR

⟨

v

⟩

and hence on its quotients.

Our goal is to extendTheorem 1.2to all characters of all finite subgroupsGofS₂

≀

S_n. To do so, we will make these pseudo-idempotents act on polynomials in bothu_iand

v

i, i.e. on the polynomial ringR

[

u

,

^v

]

. More precisely, we work in its ‘‘Segre subring’’ (see(2.4))

⨁

d⩾0

Sym^d_R(u)

⊗

Sym^d_R(v)

↪ →

Sym^2d_R (U

⊕

V)

.

^(2.5)

Note from above thatR

⟨

u

,

^v

⟩ =

T_R^•(U

⊕

V) is indeed a (S₂

≀

S_n)

×

(S₂

≀

S_n)-submodule, as is its quotient Sym^d_R(U

⊕

V) for eachd_⩾0 and hence the Segre subring. In particular, the pseudo-idempotentsE_χ^u act on it as explained in(2.12)and by fixing all factors of

v

j:

E_χ^u(u^mv^m′)

:=

E_χ^u(u^m)

·

v^m′

= χ

⁽¹⁾∑

g∈_G

χ

^(g)g−1(um)

·

v^m′

,

and similarly for theE_χ^v.

5

2.2. The main theorem and its proof, for even variables

Having defined the Segre subring and the action of the twoE_χ^•on it, we can state the promised generalization ofTheorem 1.2to all subgroupsG⩽S₂

≀

S_nand all characters – in fact, class functions –

ψ

^ofG. We begin with

ψ

a multiplicity-free character:

Theorem 2.2. Fix an integer n⩾ 1, a subgroup G ⩽ S₂

≀

S_n acting on the variables u_i (and on the

v

j) by signed permutations, and a multiplicity-free complex character

χ

of G. Then for any commutative R_χ-algebra R – see(2.1)– and any formal power series f

∈

R

[[

t

]]

(with t an indeterminate), one has:

|

G

|

E_χ^u

·

n

∏

i=₁

f(tu_i

v

i)

= |

G

|

E_χ^v

·

n

∏

i=₁

f(tu_i

v

i)

=

∑

m∈Zⁿ_⩾0

t^|m|f_m

·

E_χ^u(u^m)

·

E_χ^v(v^m)

,

^(2.6)

where the indeterminate t keeps track of theZ⩾0-grading, we use the multi-index notation m

=

(m₁

, . . . ,

^mn)

, |

m

| =

m₁

+ · · · +

m_n

,

^fm

:=

∏

i

f_mi

,

^um

:=

∏

i

u^m_iⁱ

,

^vm

:=

∏

i

v

^m_i ⁱ

,

andm⩾0is interpreted coordinatewise.

Observe that special cases of Eq.(2.6)yield Cauchy’s determinantal formula, its analogue for permanents and immanants (for the power seriesf₀(t)

=

1

/

⁽¹

−

t)), and their generalizations to arbitrary power series. E.g. for

χ

the sign and trivial representation respectively (andG

=

S_nfor n⩾2), theG-immanants have ‘‘orthogonal’’ expansions, respectively:

n

!

detf

[

tuv^T

] =

n

!

V(u)V(v) ∑

m∈Zⁿ_⩾0

t^|m|+

(

ⁿ2

)

∏ⁿ

i=₁

f_mi+_n−_i

·

s_m(u)s_m(v)

,

^(2.7)

n

!

permf

[

tuv^T

] =

∑

m∈Zⁿ_⩾0

t^|m|f_m

·

perm(u^◦m) perm(v^◦m) (2.8)

=

n

!

∑

m∈Zⁿ_⩾0,mnon-increasing

t^|m|

|

Stab_Sn(m)

|

f_m

·

m_m(u)m_m(v)

,

^(2.9)

for an arbitrary formal power seriesf(t). (Herem_m(u) denotes the monomial symmetric polyno- mial.) These equalities hold over an arbitrary unital commutative ring, and hence one can work over R

=

_Q

[

X

]

a suitable polynomial ring, cancel n

!

from all of these, then observe the ‘‘normalized’’

identity over the subringR

=

_Z

[

X

]

, and finally, specialize the variablesXto show these equalities over any unital commutative ringR.

Before proving Theorem 2.2, we make two observations. The first extends the theorem to all complex (finite-dimensional) characters ofG. Even more generally:

Corollary 2.3. Setting as inTheorem2.2. Let

ψ =

∑

χ∈ˆG_Ca_χ

χ

be any complex class function of G, and R_ψ the corresponding ring as in(2.1). Defining

E_ψ^u

:=

∑

χ∈ˆG_C

a_χE_χ^u

and similarly E^v

ψ(where we set

ψ :=

∑

χa_χ

χ

^{— using a}χ and not a_χ), we have

|

G

|

E_ψ^u

·

n

∏

i=₁

f(tu_i

v

i)

= |

G

|

E_ψ^v

·

n

∏

i=₁

f(tu_i

v

i)

=

∑

χ∈ˆG_C

a_χ ∑

m∈Zⁿ_⩾0

t^|m|f_m

·

E_χ^u(u^m)

·

E_χ^v(v^m) (2.10)

for an arbitrary R_ψ-algebra R and any f

∈

R

[[

t

]]

(and bosonic variables u_i

, v

j).

6

While this identity is more general than (2.6), it also is an immediate consequence of it, by linearity. In fact, the proof (below) of the first equality in(2.6)will also carry over verbatim to(2.10).

Our next remark explains why – forG

=

S_norS₂

≀

S_n– the above identities(2.6),(2.10)in fact hold over all rings.

Remark 2.4. Returning to(2.6), two pleasing special cases are whenG

=

S_n and G

=

S₂

≀

S_n (the typeAandBWeyl groups, respectively). In this case, Springer showed [23] that all irreducible complexG-representations can in fact be constructed overQ. In particular, all character values are integers (since they are algebraic integers and rational), and soR_χ

=

_Z. Thus, forGthe Weyl group of typeAorB, one has the immanant identity(2.6)– and hence the class function identity(2.10)– over arbitrary unital commutative rings.

We now show the above theorem. The proof has two ingredients: the first explains a key property of signed permutation matrices, when acting on symmetric functions in two sets of variables.

Lemma 2.5. Fix an integer n_⩾1and a unital commutative ring R. Given a signed permutation matrix g

=

D

· σ ∈

GL_n(R)(see the lines before(2.2)), denote its action on the u_ivia(2.2)by g^u, and similarly define g^v. These extend to actions on the ring of polynomials R

[

u₁

, . . . ,

^un

, v

1

, . . . , v

n

]

and on its Segre subring. Then for every symmetric polynomial F in n variables, and all signed permutations g, we have:

g^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

(g⁻¹)^v

·

F(u₁

v

1

, . . . ,

^un

v

n)

, ∀

F

∈

R

[ w

1

, . . . , w

n

]

^Sn

,

^g

∈

S₂

≀

S_n

.

We defer the proof as we will also show the converse result, inTheorem 2.6below.

WithLemma 2.5in hand, we can complete the proof of the theorem above.

Proof ofTheorem 2.2. We begin with an arbitrary power seriesf(t)

=

∑

m⩾0f_mt^m

∈

R

[[

t

]]

, and assert the equation

n

∏

i=₁

f(tui

v

i)

=

∑

m∈Zⁿ_⩾0

t^|m|fmu^mv^m

.

^(2.11)

Notice that Eq.(2.11)is (a) obvious, and (b) precisely the sought-for identity (Eq.(2.6)) corre- sponding to the trivial groupG

= {

1

}

.

We now return to the original setting of a general subgroupG_⩽S₂

≀

S_nacting on theu_iand on the

v

jvia signed permutations(2.2)— and a multiplicity-free (complex) character

χ

^{ofG. Working} in the Segre subring(2.5), apply the operatorsE_χ^u andE^v_χ to the above equation(2.11).²We now claim that both operations yield equal expressions on the left-hand side by reindexing. Indeed, apply Lemma 2.5with

F(u₁

v

1

, . . . ,

^un

v

n)

=

n

∏

i=₁

f(tu_i

v

i)

∈

R

[

u₁

, . . . ,

^un

, v

1

, . . . , v

n

][[

t

]] .

This yields the following calculation — e.g. in eacht-degree separately:

E_χ^v

·

n

∏

i=₁

f(tu_i

v

i)

= χ

⁽¹⁾ ∑

h=g−1∈G

χ

^{(h−1)(g−1)v}

·

F(u₁

v

1

, . . . ,

^un

v

n)

= χ

⁽¹⁾∑

g∈_G

χ

^(g)gu

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

E^u_χ

·

n

∏

i=₁

f(tu_i

v

i)

,

since

χ

^(g)

= χ

^{(g−1) for allg}

∈

G.

2 _IfG=Sn, then this precisely yields the corresponding immanant of the matrixf[tuv^T]. 7

This implies that both operations yield the same resulton the right-hand side of(2.11)as well. In particular,E_χ^u

·

E^v

χ^′

=

0 when acting on(2.11), for irreducible complex characters

χ ̸= χ

^{′, since} E_χ^u

·

E^v

χ^′

·

∑

m∈Zⁿ_⩾0

t^|m|f_mu^mv^m

=

E^u_χ

·

E^v

χ^′

·

n

∏

i=₁

f(tu_i

v

i)

=

E_χ^uE_χ^u′

·

n

∏

i=₁

f(tu_i

v

i)

,

and this vanishes by character orthogonality. This implies thatE_χ^• is also pseudo-idempotent for

χ

multiplicity-free (E_χ²

= |

G

|

E_χ), and so applying either

|

G

|

E_χ^uor

|

G

|

E_χ^vto the left-hand side of(2.11) is the same as applyingE_χ^u

·

E_χ^v:

|

G

|

E_χ^u

·

n

∏

i=₁

f(tu_i

v

i)

=

(E_χ^u)²

·

n

∏

i=₁

f(tu_i

v

i)

=

E_χ^u

·

E_χ^v

n

∏

i=₁

f(tu_i

v

i)

.

Therefore, the same observation applies to the right-hand side of (2.11) — which yields the result. _□

2.3. Larger linear groups do not work

We now explain – as promised above – why Theorem 2.2 does not extend to other finite subgroupsG⩽GL_n. The proof ofTheorem 2.2was in three steps:

(1) Lemma 2.5, which says that the actions of g^u

,

^(g−1)v

:

R

[ w

1

, . . . , w

n

]

^Sn

→

⨁

d⩾0

Sym^d_R(u)

⊗

Sym^d_R(v) are the same, where

w

i

=

u_i

v

i—ifgis a signed permutation.

(2) This implies that the actions of E_χ^u and E^v_χ are the same on the symmetric function

∏n

i=₁f(tu_i

v

i), for any multiplicity-free character

χ

^{of anyG}⩽S₂

≀

S_n. (3) Now the pseudo-idempotence ofE_χ^uandE_χ^v implies the result.

GivenTheorem 2.2, it is now natural to ask if this result can be extended fromG⩽S₂

≀

S_nto any finite matrix subgroupGofGL_n(R). In greater detail: first note that the action ofS_n (orS₂

≀

S_n) on the freeR-moduleUextends to that of matricesg

=

(m_ij)ⁿ_i,j=1

∈

GL_nvia:

g

·

n

∑

j=₁

r_ju_j

=

n

∑

i=₁

⎛

⎝

n

∑

j=₁

m_ijr_j

⎞

⎠u_i

.

^(2.12)

In turn, this GL_n-action extends to all of R

⟨

u

⟩

by multiplicativity, and then descends to aGL_n(R)- action on the quotient algebras Sym^•_R(u)

, ∧

^•_R(u). These remarks apply equally toVandU

⊕

Vin place ofU, and then one can ask ifTheorem 1.2extends to characters of a finite subgroupG _⩽ GL_n(R) that need not be contained inS₂

≀

S_n.

Here we show two negative results. The first is a converse to Step (1), and shows that over an integral domain, the conclusion ofLemma 2.5holds only for signed permutations:

Theorem 2.6. Fix a unital commutative ring R, an integer n_⩾1, and bosonic indeterminates u_i

, v

ifor i

=

1

, . . . ,

n. Given an element g

∈

GL_n(R), each of the following assertions implies the next:

(1) g is a signed permutation: g

∈

S₂

≀

S_n.

(2) g^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

(g⁻¹)^v

·

F(u₁

v

1

, . . . ,

^un

v

n) for all symmetric functions F

∈

R

[ w

1

, . . . , w

n

]

^Sn.

(3) g^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

(g⁻¹)^v

·

F(u₁

v

1

, . . . ,

^un

v

n)for the n elementary symmetric functions e_k(w)

=

∑

1⩽i1<i2<···<ik⩽n

w

i1

w

i2

· · · w

ik

,

¹⩽k⩽n

.

8

If moreover R is an integral domain, then all assertions are equivalent.

Remark 2.7. The groupS₂

≀

S_n of signed permutations affords several attractive properties over the realsR

=

_R, i.e. as a subgroup ofGL_n(R). In addition to its irreducible representations being constructible overQ(being the typeBWeyl group; seeRemark 2.4), signed permutation matrices enjoy characterizations in multiple fields. In linear algebra, they are precisely the orthogonal matrices with integer entries. In analysis, as a special case of the Banach–Lamperti theorem, they coincide with the linear isometries of thep-norms (Rⁿ

, ∥ · ∥

_p) for eachp

∈ [

1

, ∞] \ {

2

}

. Now our Theorem 2.6provides a ‘‘symmetric function’’ characterization inGL_n(R) of the signed permutation matrices – over any integral domainR– that is novel to the best of our knowledge.

Proof ofTheorem 2.6. We show a cyclic chain of implications, starting with (1)

H⇒

(2) (which wasLemma 2.5). Writeg

=

D

σ

^{, whereD}is a diagonal matrix with (i

,

^{i) entry}

ε

i

∈ {±

1

}

. Now,

g^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

D^u

σ

^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

F(

(

ε

σ(i)u_σ(i)

v

i)ⁿ_i=1)

,

whereas

(g⁻¹)^v

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

(

σ

^{−1)v(D−1)v}

·

F( (u_j

v

j)ⁿ_j=₁

)

=

F(

(u_j

ε

_j⁻¹

v

_σ⁻¹(j))ⁿ_j=₁

)

.

Now permute the arguments here via:j

= σ

(i), and use that

ε

j⁻¹

= ±

1

= ε

jtogether with the symmetry ofF, to conclude the proof.

Clearly, (2)

H⇒

(3). Now we suppose R is an integral domain, say with quotient field F, and show that (3)

H⇒

(1). We begin by recalling an observation on elementary symmetric functions that is required in this proof. Suppose an infinite fieldKcontains pairwise distinct ele- ments

w

1

, w

2

, . . . , w

nand pairwise distinct elements

w

₁^′

, w

^′₂

, . . . , w

n^′, whose elementary symmetric functions agree:

e₁(w)

= w

1

+ · · · + w

n

= w

^′₁

+ · · · + w

_n^′

=

e₁(w^′)

,

^e2(w)

=

e₂(w^′)

, . . . ,

^en(w)

=

e_n(w^′)

.

Then the polynomials (x

− w

1)

· · ·

(x

− w

n) and (x

− w

₁^′)

· · ·

(x

− w

^′n) coincide inK

[

x

]

, hence so do their sets of roots in the fieldK— i.e.,

{ w

i

:

1 ⩽ i⩽ n

} = { w

^′_i

:

1 ⩽ i ⩽ n

}

. We will apply this observation presently, with the (infinite) field beingK^′

:=

_F(u₁

, . . . ,

^un

, v

1

, . . . , v

n).

Returning to the proof, letg

=

(m_ij)ⁿ_i,j=1

∈

GL_n(R), and denote

ε :=

det(g)

∈

R^×. Also write the adjugate matrix of g as adj(g)

=

(a_ij)ⁿ_i,j=1, where a_ij equals (

−

1)^i+j times the (j

,

i)-minor of g. In particular,g⁻¹

=

(

ε

^−1aij)ⁿ_i,j=1. Now compute, forF running over the elementary symmetric polynomials innvariables:

g^u

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

F

⎛

⎝

⎛

⎝

v

i n

∑

j=₁

m_jiu_j

⎞

⎠

n

i=₁

⎞

⎠

,

(g⁻¹)^v

·

F(u₁

v

1

, . . . ,

^un

v

n)

=

F

⎛

⎝

⎛

⎝u_i

ε

⁻¹

n

∑

j=₁

a_ji

v

j

⎞

⎠

n

i=₁

⎞

⎠

.

We now apply the above observation applied toK^′; notice this is possible because asg

=

(m_ij) is invertible, no row or column is zero, and so theith argument ofg^u

·

Fis a nonzero multiple of

v

i

inF

[

u₁

, . . . ,

^un

, v

1

, . . . , v

n

]

, but not of any other

v

j. Similarly for the arguments of (g⁻¹)^v

·

F. Hence by the above observation, there exists a permutation

σ ∈

S_nsuch that

v

_σ(i) n

∑

j=₁

m_jσ(i)u_j

=

u_i

ε

⁻¹

n

∑

j=₁

a_ji

v

j

, ∀

1⩽i⩽n

.

But this is possible in the rational function fieldK^′only if the coefficients of everyu_r

v

sare equal on both sides. Thus, m_jσ(i)

=

0 whenever j

̸=

i(sog is necessarily a ‘‘generalized permutation matrix’’). Moreover, equating the coefficients ofu_i

v

σ(i)on both sides yields:

m_iσ(i)

= ε

^−1a_σ(i)i

=

det(g)⁻¹a_σ(i)i

.

^(2.13)

9