The symmetric functions catalog

An overview of symmetric functions and related topics

2021-10-28

Hook Schur polynomials

The hook Schur functions also known as supersymmetric Schur functions, are characters of the Lie superalgebra $\mathrm{gl}(m/n),$ see [Kac77]. The hook Schur functions were introduced by Berele and Regev 1983, see [BR83]. A good overview of this area can be found in the PhD thesis by E. Moens, [Moe07].

This is also the 6th variant of Schur functions considered in [Eq. 6.19, Mac92].

Supersymmetric functions

We follow the definitions in [MVdJ03]. We let $\xvec = (x_1,\dotsc,x_m)$ and $\yvec = (y_1,\dotsc,y_n).$ A function $f(\xvec,\yvec)$ is doubly symmetric if it is symmetric in each alphabet. We let $\spaceSym(\xvec/\yvec)$ denote the subspace of doubly symmetric functions with the property that substituting $x_1=t,$ $y_1=-t$ results in an expression independent of $t.$ We refer to functions in this subspace as the supersymmetric functions. For example, $x_1+x_2-y_1-y_2$ is doubly symmetric, (in $2+2$ variables), while $x_1+x_2 + y_1 + y_2$ is also supersymmetric.

The complete homogeneous supersymmetric functions are defined as

\[ \completeH_r(\xvec/\yvec) \coloneqq \sum_{j=0}^r \completeH_{j}(\xvec)\elementaryE_{r-j}(\yvec). \]

Similarly, the elementary supersymmetric functions are defined as

\[ \elementaryE_r(\xvec/\yvec) \coloneqq \sum_{j=0}^r \elementaryE_j(\xvec)\completeH_{r-j}(\yvec). \]

The supersymmetric powersum polynomials are defined as

\[ \powerSum_r(\xvec/\yvec) \coloneqq \powerSum_r(\xvec) + (-1)^r \powerSum_r(\yvec). \]

The supersymmetric monomial polynomials are defined as

\[ \monomial_\lambda(\xvec/\yvec) = \sum_{\mu\cup \nu = \lambda} \monomial_{\mu}(\xvec) \omega(\monomial_{\nu}(\yvec)). \]

All these give bases for $\spaceSym(\xvec/\yvec).$

Tableau definition

There is also a definition in terms of fillings of a Ferrers diagram of shape $\lambda.$ We fill the shape with entries

\[ 1 \lt 2 \lt \dotsb \lt k \lt 1' \lt 2' \lt \dotsb \lt l'. \]

A filling in $SST(\lambda)$ is defined as a filling of $\lambda$ with entries in the alphabet above such that rows and columns are weakly increasing. Furthermore, the unprimed entries must be strictly increasing with row index, and the primed entries must be strictly increasing with column index. The weight $x^{w_x(T)} y^{w_y(T)}$ is what you expect, keeping track of the primed and the unprimed alphabet. We have

\[ \schurHook_{\lambda/\mu}(\xvec/\yvec) \coloneqq \sum_{T \in SST(\lambda/\mu)} x^{w_x(T)} y^{w_y(T)} \]

where the expansion in the last sum is in terms of the classical schur functions.

Example.

The following tableau is an element in $SST(7,6,5,2),$ contributing with $x_1^2 x_2^2 x_3^3 y_1^4 y_2 y_3^2 y_4 y_5^2 y_6^2 y_7.$

$1$$1$$2$$3$$1'$$6'$$7'$
$2$$3$$3$$1'$$5'$$6'$ 
$1'$$2'$$3'$$4'$$5'$  
$1'$$3'$     

Weyl type formula

See 1.17 in http://igm.univ-mlv.fr/~fpsac/FPSAC02/ARTICLES/Moens.pdf

Jacobi–Trudi identity

A Jacobi–Trudi type formula for hook Schur functions was proved in [PT92], and it also follows from [Thm. 4.5, Kwo08]. The $(m,n)$-hook Schur functions are then given as

\[ \schurHook_{\lambda/\mu}(\xvec/\yvec) \coloneqq \det[ \completeH_{\lambda_i-\mu_j + j - i}(\xvec/\yvec) ]_{1\leq i,j \leq \length(\lambda)}. \]

There is also the dual version of this identity.

The functions $\schurHook_\lambda(\xvec/\yvec)$ are identically zero whenever $\lambda_{m+1} \geq n.$

Plethysm definition

The $(m,n)$-hook Schur functions can be defined in plethystic notation as

\[ \schurHook_\lambda(x_1,\dotsc,x_m/y_1,\dotsc,y_n) \coloneqq \schurS_\lambda(X - t Y) \vert_{t=-1} \]

where $X= x_1+\dotsb+x_m$ and $Y= y_1+\dotsb+y_n,$ see [Eq. (55), YR98].

Properties

The following four properties uniquely characterize the hook Schur functions, see [Mac95] and [MVdJ03].

  • (Homogeniety) The polynomial $\schurHook_{\lambda}(\xvec/\yvec)$ is a homogeneous of degree $|\lambda|.$
  • (Factorization) If $\lambda_m\geq n \geq \lambda_{m+1}$ so that $\lambda = (n^m + \tau)\cup \eta,$ then \[ \schurHook_{\lambda}(\xvec/\yvec) = \schurS_{\tau}(\xvec)\schurS_{\eta'}(\yvec) \prod_{i=1}^m\prod_{j=1}^n (x+i+y_j). \]
  • (Cancellation) We have that \[ \schurHook_{\lambda}(x_1,\dotsc,x_{m-1},t/y_1,\dotsc,y_{n-1},-t) =\schurHook_{\lambda}(x_1,\dotsc,x_{m-1}/y_1,\dotsc,y_{n-1}). \]
  • (Restriction) We have that \[ \schurHook_{\lambda}(x_1,\dotsc,x_{m-1},0/\yvec) =\schurHook_{\lambda}(x_1,\dotsc,x_{m-1}/\yvec) \] and \[ \schurHook_{\lambda}(\xvec/y_1,\dotsc,y_{n-1},0) =\schurHook_{\lambda}(\xvec/y_1,\dotsc,y_{n-1}). \]

We have the following properties of the hook Schur functions, see e.g., [YR98].

  • $\schurHook_{\lambda/\mu}(\xvec/\emptyset) = \schurS_{\lambda/\mu}(\xvec).$
  • $\schurHook_{\lambda/\mu}(\emptyset/\yvec) = \schurS_{\lambda'/\mu'}(\yvec).$
  • $\schurHook_{\lambda/\mu}(\xvec/\yvec) = \schurHook_{\lambda'/\mu'}(\yvec/\xvec).$
  • $\schurHook_{\lambda/\mu}(\xvec/ \yvec) = \sum_{\mu \subseteq \nu \subseteq \lambda} \schurS_{\nu/\mu}(\xvec) \schurS_{\lambda'/\nu'}(\yvec).$

A Weyl-type formula for $\schurHook_{\lambda}(\xvec/\yvec)$ as a quotient of determinants is given in [Eq. (1.17), MVdJ03]. This is referred to as the Sergeev–Pragacz formula, proved by Sergeev and independently in [VdJHKT90]. A skew version was proved later in 1995 by Hamel and Goulden.

In [Thm. 4.4, Rem87], a version of the hook-content formula is proved where an expression for

\[ \sum_{k,l\geq 0} t^k s^l \schurHook_\lambda(1,q,q^2,\dotsc,q^k / 1,p,p^2,\dotsc,p^l) \]

is given.

Cauchy identity

In [BR85], the following Cauchy-type identity is proved.

\[ \sum_{\lambda} \schurHook_\lambda(\xvec / \svec) \schurHook_\lambda(\yvec / \tvec) = \prod_{i,j} \frac{1+x_i t_j}{1-x_iy_j} \frac{1+y_i s_j}{1-s_i t_j} \]

A bijective proof can be found in [YR98].

Littlewood formula

M. Yang and J. Remmel [YR98] prove that

\[ \prod_{i\lt j} (1-x_i x_j) \prod_{i\gt j} (1+y_i y_j) \prod_{i, j} \frac{1}{1-x_iy_j} = 1 +\sum_{\alpha} \schurHook_\alpha(x_1,\dotsc,x_m/y_1,\dotsc,y_n) \]

where we sum over all partitions of the form

\[ \alpha = \begin{pmatrix} a_1 & a_2 & \dotsc & a_r \\ a_1+1& a_2+1& \dotsc & a_r+1 \end{pmatrix} \]

in Frobenius notation.

Generalizations

A quasisymmetric refinement is introduced in [MN18], and the symmetric hook Schur functions can be decomposed into such quasi-symmetric counterparts. That is

\[ \schurHook_\lambda(x_1,\dotsc,x_m/y_1,\dotsc,y_n) = \sum_{\alpha \sim \lambda} \schurHookQS_\alpha(x_1,\dotsc,x_m/y_1,\dotsc,y_n) \]

The quasisymmetric hook Schur functions are positive in the super Gessel fundamental basis, see [Theorem 4.2, MN18]. They conjecture that the structure constants for quasisymmetric hook Schur functions are the same as for the quasisymmetric Schur functions.


There is also a generalization in the direction of supersymmetric Schur functions indexed by composite partitions. A conjectured Jacobi–Trudi formula was presented in [Moe07] and later proved in [BDH18].

Big Schur functions

In [Shi17], K. Shigechi introduces the big Schur functions. These are closely related to Schur's P functions, and the supersymmetric Schur functions.

We consider fillings of $\lambda$ with entries in the alphabet $1' \lt 1 \lt 2' \lt 2 \lt \dotsb $ such that

  • each row has at most one marked $i$ for every $i=1,2,\dotsc$
  • each column has at most one unmarked $i,$ for every $i=1,2,\dotsc,$
  • entries in rows and columns are weakly increasing and

We let $SSShYT(\lambda)$ denote the set of such fillings.

Then, the big Schur function $\bigSchur_{\lambda}(\xvec)$ is defined as

\[ \bigSchur_{\lambda}(\xvec) = \sum_{T \in SSShYT(\lambda)} \xvec_T \]

where the weight of a tableau is obtained by treating primed entries as unprimed.

We can also realize $\bigSchur_{\lambda}(\xvec)$ as the specialization $\schurHook_{\lambda}(\xvec/\xvec).$

Example.

We have

\[ \bigSchur_{211} = 12 \monomial_{32}+4 \monomial_{41}+40 \monomial_{221}+24 \monomial_{311}+80 \monomial_{2111}+160 \monomial_{11111}. \]

Of course, there is a Jacobi–Trudi identity (also valid on the skew case):

\[ \bigSchur_{\lambda}(\xvec) = \det\left[ r_{\lambda_i -i +j}(\xvec) \right]_{1\leq i, j \leq \length(\lambda)} \]

where $r_k(\xvec) = \sum_{\mu \vdash r} 2^{\length(\mu)} \monomial_{\mu}(\xvec).$ Alternatively, $\sum_{k \geq 0} t^k r_k(\xvec) = \prod_{i} \frac{1+x_i t}{1-x_i t}.$

Factorial supersymmetric Schur polynomials

In [Def. 1.1, Mol98], Molev introduces a factorial version of supersymmetric Schur functions. They can be described via tableaux and Jacobi–Trudi identities. Molev also proves a characterization theorem and a Sergeev–Pragacz type formula, and introduces the shifted supersymmetric Schur polynomials.

In [FK20], the authors present determinant identities for skew factorial supersymmetric Schur functions.

References