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

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

- [BDH18] Nguyên Luong Thái Bình, Nguyên Thi Phuong Dung and Phùng Hô Hai. Jacobi–Trudi type formula for character of irreducible representations of $gl(m|1)$. Acta Mathematica Vietnamica, July 2018.
- [BR83] Allan Berele and Amitai Regev. Hook Young diagrams, combinatorics and representations of Lie superalgebras. Bull. Amer. Math. Soc. (N.S.), 8(2):337–339, 1983.
- [BR85] A. Berele and Jeffrey B. Remmel. Hook flag characters and their combinatorics. Journal of Pure and Applied Algebra, 35:225–245, 1985.
- [FK20] Angèle M. Foley and Ronald C. King. Factorial supersymmetric skew Schur functions and ninth variation determinantal identities. arXiv e-prints, 2020.
- [Kac77] V. G. Kac. Lie superalgebras. Advances in Mathematics, 26(1):8–96, October 1977.
- [Kwo08] Jae-Hoon Kwon. Rational semistandard tableaux and character formula for the Lie superalgebra $gl{\infty|\infty}$. Advances in Mathematics, 217(2):713–739, January 2008.
- [Mac92] I. G. Macdonald. Schur functions: Theme and variations. Séminaire Lotharingien de Combinatoire [electronic only], 28:5–39, 1992.
- [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
- [MN18] Sarah K. Mason and Elizabeth Niese. Quasisymmetric {$(k,l)$}-hook Schur functions. Annals of Combinatorics, 22(1):167–199, February 2018.
- [Moe07] Els Moens. Supersymmetric Schur functions and Lie superalgebra representations. Ghent University. 2007.
- [Mol98] Alexander Molev. Factorial supersymmetric schur functions and super Capelli identities. In Kirillov's seminar on representation theory. Amer. Math. Soc., Transl. Ser. 2, 1998.
- [MVdJ03] Els Moens and Joris Van der Jeugt. A determinantal formula for supersymmetric Schur polynomials. Journal of Algebraic Combinatorics, 17(3):283–307, 2003.
- [PT92] Piotr Pragacz and Anders Thorup. On a Jacobi–Trudi identity for supersymmetric polynomials. Advances in Mathematics, 95(1):8–17, September 1992.
- [Rem87] J. B. Remmel. Permutation statistics and $(k,l)$-hook Schur functions. Discrete Mathematics, 67(3):271–298, December 1987.
- [Shi17] Keiichi Shigechi. Shifted tableaux and products of Schur's symmetric functions. arXiv e-prints, 2017.
- [VdJHKT90] Joris Van der Jeugt, J. W. B. Hughes, Ronald C. King and Jean Thierry-Mieg. Character formulas for irreducible modules of the Lie superalgebras $sl(m/n)$. Journal of Mathematical Physics, 31(9):2278–2304, September 1990.
- [YR98] M. Yang and J. B. Remmel. Hook-Schur functions analogues of Littlewood's identities and their bijective proofs. European Journal of Combinatorics, 19(2):257–272, February 1998.