Schur Q-functions and Capelli eigenvalues for \mathfrak{q}(n)

# Schur Q-functions and the Capelli eigenvalue problem for the Lie superalgebra q(n)

Alexander Alldridge Mathematisches Institut
Mathematisch-Naturwissenschaftliche Fakultät
Universität zu Köln
Weyertal 86–90
50931 Köln
Germany
Department of Mathematics
University of California at Berkeley
969 Evans Hall
Berkeley, CA 94720
USA
https://aalldridge.github.io
Siddhartha Sahi Department of Mathematics
Rutgers University
110 Frelinghuysen Rd
Piscataway, NJ 08854-8019
United States of America
and  Hadi Salmasian Department of Mathematics and Statistics
University of Ottawa
585 King Edward Ave
Ottawa, Ontario
###### Abstract.

Let , where denotes the queer Lie superalgebra. The associative superalgebra of type has a left and right action of , and hence is equipped with a canonical -module structure. We consider a distinguished basis of the algebra of -invariant super-polynomial differential operators on , which is indexed by strict partitions of length at most . We show that the spectrum of the operator , when it acts on the algebra of super-polynomials on , is given by the factorial Schur -functions of Okounkov and Ivanov. As an application, we show that the radial projections of the spherical super-polynomials (corresponding to the diagonal symmetric pair , where ) of irreducible -submodules of are the classical Schur -functions. As a further application, we compute the Harish-Chandra images of the Nazarov basis of the centre of .

###### Key words and phrases:
Capelli identity, queer Lie superalgebra, Schur -function
###### 2010 Mathematics Subject Classification:
Primary 17B10; Secondary 17B60, 58A50
Alexander Alldridge gratefully acknowledges support by the German Research Council (Deutsche Forschungsgemeinschaft DFG), grant nos. AL 698/3-1 and ZI 513/2-1, and the Institutional Strategy of the University of Cologne in the Excellence Initiative. The research of Siddhartha Sahi was partially supported by a Simons Foundation grant (509766) and of Hadi Salmasian by an NSERC Discovery Grant (RGPIN-2013-355464). This work was initiated during the Workshop on Hecke Algebras and Lie Theory, which was held at the University of Ottawa. Hadi Salmasian and Siddhartha Sahi thank the National Science Foundation (DMS-162350), the Fields Institute, and the University of Ottawa for funding this workshop

## 1. Introduction

Let be a Hermitian symmetric space of tube type. The Shilov boundary of is of the form , where is the Siegel parabolic subgroup and is a symmetric subgroup of both and . Let , , and be the complexified Lie algebras of , , and , respectively. We set and regard as an -module. In this setting, has the structure of a simple Jordan algebra.

The polynomial algebra decomposes as the multiplicity-free direct sum of simple -modules , indexed naturally by partitions . In this situation one has canonical invariant “Capelli” differential operators of the form , where is the Jordan norm polynomial. The spectrum of these operators was computed by Kostant and Sahi [ks91, ks93], and a close connection with reducibility and composition factors of degenerate principal series was established by Sahi [SahiCompositio, SahiCrelle, SahiSh].

Sahi showed [s94] that the decomposition of in fact yields a distinguished basis , called the Capelli basis, of the subalgebra of -invariant elements of the algebra of differential operators on with polynomial coefficients. Moreover, there is a polynomial , uniquely characterized by its degree, symmetry, and vanishing properties, such that acts on each simple summand by the scalar . The problem of characterizing the spectrum of the operators is referred to as the Capelli eigenvalue problem.

In fact, Sahi [s94] introduced a universal multi-parameter family of inhomogeneous polynomials that serve as a common generalization of the spectral polynomials across all Hermitian symmetric spaces of rank . Later, Knop and Sahi [ks96] studied a one-parameter subfamily of these polynomials, which already contains all the spectral polynomials. They showed that these polynomials are eigenfunctions of a class of difference operators extending the Debiard–Sekiguchi differential operators. It follows that the top degree terms of the Knop–Sahi polynomials are Jack polynomials, which for special choices of the parameter become spherical functions.

These polynomials were later studied from a different point of view by Okounkov and Olshanski, who referred to them as shifted Jack polynomials.

Subsequently, supersymmetric analogs of the Knop–Sahi shifted Jack polynomials were constructed by Sergeev and Veselov in [SerVes]. More recently, two of us (Sahi and Salmasian [sahi-salmasian]) have extended this circle of ideas to the setting of the triples of the form

 (1.1) (gl(m|n)×gl(m|n),gl(m|n),Matm|n(C)),(gl(m|2n),osp(m|2n),S2(Cm|2n)).

In each of these situations one has, once again, a canonical Capelli basis of differential operators, and [sahi-salmasian] establishes a precise connection to the abstract Capelli problem of Howe and Umeda [HoweUmeda]. It is further shown in Ref. [sahi-salmasian] that the spectrum of the Capelli basis is given by specialisations of super analogues of Knop–Sahi polynomials, defined earlier by Sergeev and Veselov [SerVes]. In the case of the triple , these results follow from earlier work of Molev [MolevFSY], however the case is harder and requires new ideas.

The Lie superalgebras and are examples of basic classical Lie superalgebras. Such an algebra admits an even non-degenerate invariant bilinear form and an even Cartan subalgebra, and many results for ordinary Lie algebras extend to this setting, see for instance Ref. [as], where spherical representations for the corresponding symmetric pairs are studied. In this paper, we show that the ideas of Ref. [sahi-salmasian] can actually be extended to non-basic Lie superalgebras. More precisely, we consider the case of the queer Lie superalgebra , usually defined as the subalgebra of of matrices commuting with an odd involution [KacLSA]. For the present purposes, it is convenient to work with a slightly different realization of , which we describe below.

Let be the -algebra generated by an odd element , with ; thus as a superspace, . Let be the associative superalgebra of matrices with entries in . Then is the associative superalgebra of type , and is isomorphic to regarded as a Lie superalgebra via the graded commutator

 [x,y]\coloneqqxy−(−1)|x||y|yx.

In fact is also a Jordan superalgebra via the graded anticommutator, and an -bimodule via left and right multiplication. This bimodule structure induces a -module structure on .

In this paper, we consider the Capelli eigenvalue problem for the “diagonal” triple

 (1.2) (l,m,V)\coloneqq(q(n)×q(n),q(n),A).

We establish a close connection with the Schur -functions and their inhomogeneous analogues, the factorial Schur -functions , which were originally defined by Okounkov and studied by Ivanov [ivanov97]. Our main results are as follows. From Ref. [cw], it is known that the space of super-polynomials on decomposes as a multiplicity-free direct sum of certain -modules , which are parametrised by strict partitions of length at most . It follows that decomposes as a direct sum of the contragredient -modules . In Section 4.3, we describe a certain even linear slice to the -orbits on . If is an -invariant super-polynomial on , then it is uniquely determined by its restriction to . This restriction is an ordinary polynomial, and we call it the -radial part of

###### Theorem 1.1.

For every , the -module contains an -spherical super-polynomial , which is unique up to a scalar multiple. Moreover, up to a scalar, the -radial part of is the Schur -function .

This is proved in Theorem 4.5 below. Now consider the algebra of polynomial coefficient differential operators on . It has an -module decomposition

 PD(V)≅⨁λ,μVμ⊗V∗λ≅⨁λ,μHom–––––l(Vλ,Vμ),

and we write for the differential operator corresponding to the identity map . The are the Capelli operators, and they form a basis for the -invariant differential operators acting on . The operator acts on each irreducible component of by a scalar eigenvalue .

###### Theorem 1.2.

The eigenvalues of the Capelli operator are given by the factorial Schur -function . More precisely, for all , we have

 cλ(μ)=Q∗λ(μ)Q∗λ(λ).

In fact, we prove Theorem 1.2 first (see Theorem 3.8 below) and then use it to prove Theorem 1.1.

Compared to the cases considered in Equation (1.1), the situation in Equation (1.2) is more complicated. First, since the Cartan subalgebra of is not purely even, the highest weight space of an irreducible finite dimensional -module is not necessarily one-dimensional. Second, unlike the basic classical cases, the tensor product of two irreducible -modules is not necessarily an irreducible -module, and sometimes decomposes as a direct sum of two modules which are isomorphic up to parity change. Third, the -spherical vectors in are purely odd, whereas the -spherical vectors in are purely even. These issues add to the difficulties that arise in the proofs in the case of the symmetric pair in Equation (1.2).

In [sergeev]*Theorem 3, Sergeev introduced the -analogue of the Harish-Chandra isomorphism

 (1.3) η:Z(q(n))⟶P(h¯0),

where denotes the even part of the Cartan subalgebra of . Theorem 1.2 can be reformulated in terms of the map , as follows. The image of can be naturally identified with the space of -variable -symmetric polynomials (see Section 3.2). We denote the actions of the first and second factors of on by and , respectively. Since the typical “-shift” for the Sergeev–Harish-Chandra isomorphism is equal to zero, we obtain the following reformulation of Theorem 1.2.

###### Theorem 1.3.

For every Capelli operator , there exists a unique central element such that . Furthermore,

 η(zλ)(μ)=Q∗λ(μ)Q∗λ(λ).

The setting of the present paper was also considered by Nazarov, who constructed [nazarov]*Eq. (4.7) a family of invariant differential operators using characters of the Sergeev algebra [sergeev]. Nazarov also defined [nazarov]*Eq. (4.6) certain explicit “Capelli” elements in , and proved [nazarov]*Cor. 4.6 that , where is the left action of on .

Although our operators and central elements are different from the and defined by Nazarov, one can make an a posteriori connection using our Proposition 3.6 below. This allows us to compute the Harish-Chandra image of Nazarov’s central elements . The following result follows immediately from Theorem 4.9.

###### Theorem 1.4.

The Harish-Chandra image of the operator is given by

 η(Cλ)(μ)=kλQ∗λ(−μ),

where

 kλ\coloneqq(−1)|λ|λ1!⋯λℓ(λ)!∏1⩽i

We would like to mention that the polynomials occur in a further different scenario related to the Lie superalgebra . In [sergeevv]*Theorem 1.7, Sergeev showed that the radial parts of the bispherical matrix coefficients on with respect to the diagonal and twisted-diagonal embeddings of in , are Schur -polynomials. It will be interesting to explore possible connections between our work and Sergeev’s result.

We remark that it is possible to extend the results of the present paper and of Ref. [sahi-salmasian] to the common setting of multiplicity-free actions on Jordan superalgebras. This will be established in a forthcoming paper [SahiSalSer]. In addition, recently Sahi and Salmasian [sahi-salmasian-ENS] constructed quadratic analogues of Capelli operators on Grassmannian manifolds by lifting the Capelli basis of [s94] via a double fibration. In the near future they plan to consider the analogous problem in the super setting.

We conclude this introduction with a brief synopsis of our paper. In Section 2, we realise the Lie superalgebras relevant to us in terms of supermatrices. In Section 3, we identify the action of on , construct the Capelli basis, and determine the eigenvalue polynomials (Theorem 3.8). Finally, in Section 4, we study the open orbits in and , show the existence of -invariant functionals for the simple summands of , and prove that the spherical polynomials thus defined are the classical Schur -functions (Theorem 4.5).

Acknowledgements. Alexander Alldridge wishes to thank the University of Ottawa, the Institute for Theoretical Physics at the University of Cologne, and the Department of Mathematics of the University of California at Berkeley for their hospitality during the preparation of this article.

The authors thank Vera Serganova and Weiqiang Wang for stimulating and fruitful conversations during the workshop, which paved the way for the present article. We also thank Alexander Sergeev for bringing Ref. [sergeevv] to our attention.

## 2. Lie superalgebras

The triple given in Equation (1.2) can be embedded inside the Lie superalgebra , which can be further embedded inside . This provides a concrete realisation which allows us to express the Lie superalgebras of interest as matrices. In order to describe it, it will be convenient to consider three commuting involutions of the algebra . To this end, first we equip the space of complex matrices with a Lie superalgebra structure isomorphic to . Instead of supermatrices in standard format, we prefer to consider those of the shape

 (2.1) x=(ABCD),A,B,C,D∈gl(n|n).

Equipped with the signed matrix commutator, the space of such matrices forms a Lie superalgebra isomorphic to , the isomorphism being given by conjugation by the matrix

 ˜I\coloneqq⎛⎜ ⎜ ⎜⎝I00000I00I00000I⎞⎟ ⎟ ⎟⎠,

where denotes the identity matrix. Next let , and be three involutions on , given respectively by conjugation by the matrices

 Σ\coloneqq⎛⎜ ⎜ ⎜⎝I0000I0000−I0000−I⎞⎟ ⎟ ⎟⎠ , Φ\coloneqq⎛⎜ ⎜ ⎜⎝0I00I000000I00I0⎞⎟ ⎟ ⎟⎠ , Θ\coloneqq⎛⎜ ⎜ ⎜⎝00I0000II0000I00⎞⎟ ⎟ ⎟⎠.

It is straightforward to verify that , , and . Hence, the involutions , and commute with each other.

### 2.1. The involution σ

The subspace of fixed points of equals with elements of the form

 (A00D) where A,D∈gl(n|n).

The subspace of fixed points of consists of matrices of the form

 (0BC0) where B,C∈gl(n|n).

Thus, as a super-vector space, where are respectively the spaces of matrices of the form

 (0B00) and (00C0)

with . In fact, are the -eigenspaces of , where we think of as an element of . Therefore, are abelian subalgebas of , and together with , they form a -grading of . The action of on is given explicitly by

 (2.2) [x,v]=(0AB−(−1)|B||D|BD00),

for all homogeneous

 x=(A00D)∈gσ,v=(0B00)∈g−σ+.

In what follows, we set and identify it as a super-vector space with , via the map

 (2.3) gl(n|n)⟶U:B⟼(0B00).

### 2.2. The involution φ

The involution is induced by the parity reversing automorphism of , and therefore the subalgebra of fixed points of is isomorphic to . It consists of all as in Equation (2.1) such that the blocks . Furthermore,

 l\coloneqqgφ,σ≅q(n)×q(n).

We define . Then it is clear that

 [l,V]⊆gφ∩[gσ,g−σ+]⊆gφ∩g−σ+=V.

The restriction of the map defined in Equation (2.3) yields an identification of with a subspace of which carries the structure of .

### 2.3. The involution θ

The algebra of fixed points of both and is isomorphic to , and realised by supermatrices

 (2.4) (A00A),A∈gl(n|n).

From Equation (2.2), it follows that the action of on is precisely the adjoint action of . Furthermore,

 m\coloneqqlθ=gφ,σ,θ≅q(n).

It is realised by supermatrices of the form as in Equation (2.4) where in addition . Moreover, the action of on is precisely the adjoint action of . For the following lemma, let be the element correponding to the matrix

 (0I2n×2n00),

and set

 s\coloneqql−θ=gφ,σ,−θ.
###### Lemma 2.1.

Let be the linear map defined by .

1. The map is a surjection onto with kernel . Its restriction to is a linear isomorphism.

2. The restriction of to is a surjection onto with kernel . Its restriction to is a linear isomorphism onto .

3. Equipped with the binary operation for (respectively, ) the super-vector space (respectively, ) becomes a Jordan superalgebra.

###### Proof.

Parts (i) and (ii) follow from straightforward calculations. For (iii), it suffices to prove the statement for . Note that according to Equation (2.2), we have

 x⋅e=(02A00),∀x=(A00−A)∈gσ,−θ.

Thus, a direct calculation shows that up to normalization, coincides for with the super-anticommutator . ∎

## 3. The eigenvalue polynomials

### 3.1. The action of l on polynomials

Let denote the superalgebra of super-polynomials on . Recall that is by definition equal to . As acts on , we obtain an induced locally finite -action on . Similar statements apply to . We shall identify this action in terms of differential operators. To that end, we consider the complex supermanifold , defined as the locally ringed space with underlying topological space and sheaf of superfunctions , where denotes the sheaf of holomorphic functions. There is a natural inclusion

 V∗⟶Γ(OA(V))

( denoting global sections), allowing us to identify linear forms on with certain superfunctions on . In particular, is a subsuperalgebra of .

Recall that on a supermanifold , the vector fields on , defined on an open set of the underlying topological space, are defined to be the superderivations of [ahw-orbits]*Definition 4.1. For any homogeneous basis of , the dual basis is a coordinate system on , and [ahw-orbits]*Proposition 4.5 there are unique (and globally defined) vector fields on of parity , determined by

 ∂∂xa(xb)=δab,∀a,b.

The linear action of on determines, for , vector fields on by

 (3.1) av(xb)=−v⋅xb,∀b.

The sign stems from the fact that these are the fundamental vector fields for a Lie supergroup action on , as we shall see later. By construction, for any , the action of on coincides with the action of defined in the first paragraph of this subsection.

We now make this action explicit. Let be the standard basis of . A homogeneous basis of the super-vector space is determined by

 ukℓ\coloneqq(Ekℓ00Ekℓ),ξkℓ\coloneqq(0EkℓEkℓ0),

where and are the usual elementary matrices. Then

 ukℓ(ej) =δℓjek, ukℓ(εj) =δℓjεk, ξkℓ(ej) =δℓjεk, ξkℓ(εj) =δℓjek.

Let be the dual basis of . This determines vector fields

 ∂∂ukℓ,∂∂ξkℓ

on , by the recipe given above.

Moreover, let and , respectively, be the copies of in the first and second factor of . By Equation (2.2), as a module over the second factor, is isomorphic to where acts on in the standard way. Hence, the second factor acts on as on :

 bkℓ(upq) =δℓqupk bkℓ(ξpq) =δℓqξpk, βkℓ(upq) =δℓqξpk βkℓ(ξpq) =δℓqupk.

It follows that

 (3.2) −abkℓ=n∑p=1upk∂∂upℓ+ξpk∂∂ξpℓ,−aβkℓ=n∑p=1ξpk∂∂upℓ+upk∂∂ξpℓ.

Similarly, the first factor of acts on as on , and hence on as on . Reasoning as above, this implies

 (3.3) −aakℓ=n∑p=1upℓ∂∂upk+ξpℓ∂∂ξpk,−aαkℓ=n∑p=1−ξpℓ∂∂upk+upℓ∂∂ξpk.

We will presently decompose the -module . To that end, we introduce a labelling set. Let be the set of partitions, that is, of all finite sequences of non-negative integers such that . Here, we identify with any partition obtained from by appending a finite number of zeros at its tail. If can be written in the form where , then we say has length . Let be the set of partitions of length . We also set if . A partition of length is called strict if . The set of all strict partitions will be denoted by , and we write for the set of strict partitions of length at most ; that is, .

For every strict partition such that , let be the -highest weight module with highest weight , where the are the standard characters of the even part of the Cartan subalgebra of . For every strict partition , set when is even, and otherwise. We now define an -module as follows. It is shown in [cw]*Section 2 that, as an -module, the exterior tensor product is irreducible when , and decomposes into a direct sum of two irreducible isomorphic -modules (via an odd map) if . Following the notation of Ref. [cw], we set

 Vλ\coloneqq12δ(λ)(Fλ)∗⊠Fλ),

that is, we take to be the irreducible component of that appears in the decomposition of the super-polynomial algebra over the natural -module (see Proposition 3.1). The -module is always of type , that is, it is irreducible as an ungraded representation. It follows that in the -graded sense,

 (3.4) Hom–––––l(Vλ,Vμ)=δλμ⋅C,

where a priori, denotes the set of all -equivariant linear maps (of any parity). In particular, all non-zero -equivariant endomorphisms of are even.

###### Proposition 3.1.

Under the action of , decomposes as the multiplicity-free direct sum of simple modules , where ranges over elements of which satisfy .

###### Proof.

Recall that as -modules. The proposition follows from the description of the actions of the left and right copies of on given above, and the results of [cw]*Section 3. ∎

### 3.2. Invariant polynomial differential operators

On a complex supermanifold , the differential operators on defined on an open set of the underlying topological space are generated as a subsuperalgebra of the -linear endomorphisms of by vector fields and functions. This gives a -algebra sheaf that is an -bisupermodule, filtered by order.

Here, a differential operator is of order if it can be expressed as a product of some functions and at most vector fields. A differential operator of order is uniquely determined by its action on monomials of order in some given system of coordinate functions. This follows in the usual way from the Hadamard Lemma [leites]*Lemma 2.1.8 and implies that is locally free as a left -supermodule.

For the supermanifold , we have a -superalgebra map

 ∂:S(V)⟶Γ(DA(V)).

It is determined by the linear map which sends any homogeneous to the unique vector field such that

 (3.5) ∂(v)(μ)=(−1)|μ||v|μ(v)

for all homogeneous . If is a homogeneous basis of , then

 ∂(xa)=(−1)|xa|∂∂xa.

The image of , denoted by , is the superalgebra of constant-coefficient differential operators on . The map

 (3.6) P(V)⊗D(V)⟶Γ(DA(V)):p⊗D⟼pD

is an isomorphism onto a -submodule of denoted by . Indeed, is a subsuperalgebra, the algebra of polynomial differential operators. As acts by linear vector fields, we have the bracket relation

 [−ax,p]=−ax(p)=x⋅p,∀x∈l,p∈P(V).

The bilinear form

 (3.7) Pk(V)⊗Dk(V)⟶C:p⊗D⟼Dp

is a non-degenerate pairing which is -equivariant, and therefore results in a canonical -module isomorphism . Hence the following corollary to Proposition 3.1 holds.

###### Corollary 3.2.

The space of -invariant polynomial differential operators decomposes as follows:

 PD(V)l=⨁λ∈Λn>0(Vλ⊗(Vλ)∗)l.

There is a natural -equivariant isomorphism , so the identity element of determines an even -invariant polynomial differential operator

 Dλ∈PD(V)l.

That is, if is a homogeneous basis of and is its dual basis for , then

 (3.8) Dλ\coloneqq∑jpjDj.

Schur’s Lemma implies that for every , there is a complex scalar such that acts by the scalar on .

###### Corollary 3.3.

The operators , where ranges over , form a basis of the space of -invariant differential operators. Moreover, , while whenever and .

###### Proof.

The first statement follows from Corollary 3.2 and Equation (3.4). Since the order of the operator is not , it vanishes on for . Next assume . If and does not vanish on , then the restriction of the bilinear form (3.7) to will be a nonzero -equivariant form, hence , which is a contradiction.

It remains to compute the action of on . Let and denote the dual bases of and . Then for every . ∎

To determine (which will be done in the next subsection), we first need to see that it extends to a -symmetric polynomial. To that end, let denote the ring of -symmetric polynomials, that is, -variable symmetric polynomials such that does not depend on . (For , the latter condition is vacuous.)

###### Proposition 3.4.

For all , there exists a polynomial of degree at most such that for all .

###### Proof.

Recall that and , respectively, denote the actions of the first and second factor of on . As is the multiplicity-free direct sum of simple modules of the universal enveloping algebra , it follows that and are mutual commutants in (this double commutant property is also mentioned in [cw]). In particular, we have

 PD(V)l⊆R(U(q(n))).

As is faithful, it follows that in fact

 PD(V)l=R(Z(q(n))),

where denotes the centre of . The latter statement also follows from the explicit construction of the Capelli operators in the work of Nazarov [nazarov].

Furthermore, the simple module occuring in the decomposition of is contained in the external tensor product . By Sergeev’s Harish-Chandra isomorphism for [sergeev]*Theorem 3 (see also [cwbook]*Theorem 2.46), for any , there exists a -symmetric polynomial such that for all , acts on by .

Fix . Then the order of is . As is faithful, there is a unique such that . Since acts by linear vector fields, lies in the -th part of the standard increasing filtration of . Then has degree at most , see [sergeev, cwbook]. The assertion follows. ∎

###### Definition 3.5.

We call the eigenvalue polynomial of for .

### 3.3. Schur Q-functions

Our next goal is to identify the eigenvalue polynomials . We first recall the definitions of certain elements of , called the -functions of Schur, and their shifted analogues, the factorial -functions originally defined by Okounkov, see Ref. [ivanov97].

Given a sequence of complex numbers, we define for any non-negative integer the th generalized power of by

 (x∣a)k\coloneqqk∏i=1(x−ai),

where we set . For every , we set

 Fλ(x1,…,xn∣a) \coloneqqℓ(λ)∏i=1(x∣a)λi∏i⩽ℓ(λ)i

We now define

 Qλ(x1,…,xn∣a) \coloneqq2ℓ(λ)(n−ℓ(λ))!∑σ∈SnFλ(xσ(1),…,xσ(n)∣a).

We remark that can be expressed as a ratio of an antisymmetric polynomial by the Vandermonde polynomial, and therefore it is also a polynomial. Two special cases of interest are

 Qλ(x1,…,xn)\coloneqqQλ(x1,…,xn∣0–),

where , and

 Q∗λ(x1,…,xn)\coloneqqQλ(x1,…,xn∣δ),

where , called, respectively, the Schur -function, and the factorial Schur -function.

###### Proposition 3.6.

Let .

1. We have . Furthermore, is homogeneous of degree and

 deg(Q∗λ−Qλ)<|λ|.
2. Both and are bases for the vector space .

3. We have for every such that , ; moreover,

 Q∗λ(λ)=Qλ(λ)=H(λ)\coloneqqλ!∏i

where , and it is understood that the product extends only up to .

4. The unique element of of degree at most which satisfies Equation (iii) is precisely .

###### Proof.

Parts (i)–(iii) follow from [ivanov97]*§ 1 (see also [Stembridge]*§ 6). Thus, we only sketch the argument for item (iv). Set and let consist of the polynomials of degree at most . From item (ii), it follows that , where

 Λn>0(k)\coloneqq{λ∈Λn>0∣∣|λ|⩽k}.

For every , we consider the linear functional defined by

 εμ(f)\coloneqqf(μ),∀f∈Γn.

Let the order on be defined by

 μ⊆νif and only if∀j:μj⩽νj.

We choose a total order on such that if either or and . Then by [ivanov97]*Proposition 1.16, in terms of , the matrix

 (εμ(Q∗ν))μ,ν∈Λn>0(k)

is triangular with no zeros on the diagonal, and therefore, invertible. It follows that the linear system

 εμ(f)={H(λ)if μ=λ,0if μ≠λ.

has a unique solution in . In view of item (iii), this solution is . ∎

###### Remark 3.7.

The polynomials also appear in Ref. [Macdonald]. They were initially introduced by Schur in connection with the projective representations of the symmetric group. According to [ivanov97]*Proposition 1.16, the vanishing property of Proposition 3.6 (iii) holds in a stronger form: whenever .

###### Theorem 3.8.

Let for be the eigenvalue polynomial of Proposition 3.4. Then

 q∗λ=1λ!∏i
###### Proof.

By Proposition 3.4, satisfies the same degree, symmetry, and vanishing properties as , see Proposition 3.6 (iii). The statement now follows from Proposition 3.6 (iv). ∎

###### Remark 3.9.

Nazarov [nazarov]*Proposition 4.8 constructs certain Capelli elements