Hook Schur polynomials

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#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 A. Berele and A. Regev in 1983; see [BR83]. A good overview of this area can be found in the PhD thesis by E. M. Moens [Moe07].

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

S. Naprienko introduces free fermionic Schur functions $\schurS_{\lambda/\mu;\avec,\bvec}(\xvec/\yvec)$ [Nap24]. These depend on two alphabets and two sequences of parameters, and specialize to several familiar families, including factorial, supersymmetric, and dual Schur functions. The construction is based on a free-fermionic six-vertex model and gives supersymmetric Cauchy identities, determinantal formulas, and flagged formulas. S. Iwao gives a free-fermionic presentation of multi-Schur functions [Iwa23]. This family generalizes supersymmetric Schur functions, flagged Schur functions, and refined dual Grothendieck functions, and Iwao gives a method for expanding multi-Schur functions in refined dual Grothendieck polynomials. H. Riedtmann proves overlap identities for Littlewood–Schur functions [Rie18]. These are closely related to the supersymmetric, or hook, Schur functions; the identities are obtained from the Moens–Van der Jeugt determinantal formula by Laplace expansion.

#Supersymmetric functions

We follow the definitions in [MVdJ03]. 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 power-sum polynomials are defined as \[\powerSum_r(\xvec/\yvec) \coloneqq \powerSum_r(\xvec) + (-1)^{r-1} \powerSum_r(\yvec).\] 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 down columns, and the primed entries must be strictly increasing along rows. The weight $x^{w_x(T)} y^{w_y(T)}$ keeps track of the primed and the unprimed alphabets. 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 (A small hook Schur polynomial).

For the alphabet $1\lt2\lt1'$ and shape $(3,1),$ the two hook tableaux

$1 $ $ 1 $ $ 1'$ $ 2$ $1 $ $ 1 $ $ 2$ $ 1'$

both contribute the monomial $x_1^2x_2y_1.$ Enumerating all hook tableaux of this shape gives \[\begin{aligned} \schurHook_{(3,1)}(x_1,x_2/y_1) ={}&x_1^3x_2+x_1^2x_2^2+x_1x_2^3+x_1^3y_1 +2x_1^2x_2y_1 \\ &\quad +2x_1x_2^2y_1+x_2^3y_1 +x_1^2y_1^2+x_1x_2y_1^2+x_2^2y_1^2. \end{aligned}\]

Example

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

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

#Weyl type formula

See [Eq. (1.17), MVdJ03].

#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 $\lambda_{m+1} \geq n.$

A. Adilzhan and D. Yeliussizov give Hamel–Goulden type ribbon-decomposition determinant formulas for flagged supersymmetric Schur functions [AY25].

#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].

(Homogeneity) The polynomial $\schurHook_{\lambda}(\xvec/\yvec)$ is 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].

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

N. Hage gives a super Littlewood–Richardson type rule for super Schur functions over signed alphabets [Hag25]. The coefficients are interpreted using super Young tableaux. D. T. Hiep and K.-H. Nguyen-Dang prove that supersymmetric Schur polynomials have the saturated Newton polytope property [HN25]. Their proof encodes the tableau support by a totally unimodular polyhedron and then applies the Hoffman–Kruskal integrality criterion.

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 A. Sergeev and independently in [VdJH+90]. A skew version was proved later in 1995 by A. Hamel and I. Goulden.

M. Khovanov explains how an extension of the Robert–Wagner foam evaluation to overlapping theta-foams recovers the Sergeev–Pragacz formula for supersymmetric Schur functions [Kho20]. The same framework also connects Schur and supersymmetric Schur determinants with Hankel and Toeplitz matrices arising from two-dimensional topological theories.

Thom polynomials give another geometric setting where Schur functions in a difference of alphabets occur naturally: the Chern classes are those of a virtual bundle, and Schur expansions of Thom polynomials often have strong positivity properties.

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 quasisymmetric 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 [Thm. 4.2, MN18]. They conjecture that the structure constants for quasisymmetric hook Schur functions are the same as for the quasisymmetric Schur functions.

S. Fishel, J. Gatica, L. Lapointe, and Maria Elena Pinto study the fundamental quasisymmetric functions in superspace [FGLP25]. They describe the coproduct, product, and antipode on this fundamental basis. They also extend Gessel’s expansion of Schur functions into fundamental quasisymmetric functions to skew Schur functions in superspace. S. Fishel, L. Lapointe, and Maria Elena Pinto establish Hopf-algebra structures on symmetric and quasisymmetric functions in superspace [FLP19]. Here superspace means adjoining anticommuting variables to the ordinary commuting variables. J. Lentfer proves the diagonal supersymmetry conjecture for coinvariant rings [Len25]. More generally, bosonic-fermionic coinvariant rings for finite groups carry a $U(\mathfrak{gl}(k|j))\otimes \setC[G]$-module structure, and their character series expand in super Schur functions with universal coefficients.

D. Galakhov, A. Morozov, and N. Tselousov give a combinatorial construction of supersymmetric Schur, Jack, and Macdonald polynomial families indexed by super-Young diagrams [GMT24]. Their super-Macdonald polynomials recover the ordinary Macdonald polynomials in the purely even case and carry a representation of a super-algebra analogue of the Ding–Iohara–Miki algebra.

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.

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 in the skew case): \[\bigSchur_{\lambda}(\xvec) = \det\left[ r_{\lambda_i -i +j}(\xvec) \right]_{1\leq i, j \leq \length(\lambda)}\] where \[r_k(\xvec) = \sum_{\mu \vdash k} 2^{\length(\mu)} \monomial_{\mu}(\xvec).\] Alternatively, \[\sum_{k \geq 0} t^k r_k(\xvec) = \prod_{i} \frac{1+x_i t}{1-x_i t}.\]

Shigechi also gives expansions of skew big Schur functions in products of skew Schur $P$-functions. Equivalently, after the standard powers of $2$ relating $P$- and Schur $Q$-functions, these can be written in products of skew Schur $Q$-functions. The same work gives determinant and Pfaffian Giambelli formulae for skew big Schur functions.

#Factorial supersymmetric Schur polynomials

In [Def. 1.1, Mol98], A. 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. S. Okada introduces generalized Schur $P$- and $Q$-functions associated with a polynomial sequence [Oka19]. These are recorded as generalized Schur $P$- and $Q$-functions. They are Macdonald’s ninth variation for the $P$/$Q$ setting, and include ordinary Schur $P$/$Q$-functions, factorial $P$/$Q$-functions, and the $t=-1$ specializations of Hall–Littlewood functions for classical root systems. A. M. Foley and R. C. King prove determinantal and Pfaffian identities for ninth-variation skew Schur functions and $Q$-functions [FK21]. Their approach uses non-intersecting lattice paths and gives ninth-variation analogues of outside decomposition identities. W. Takeda and Y. Yamasaki prove quadratic relations for ninth variations of Schur functions and apply them to Schur multiple zeta functions [TY25]. M. Goltsblat extends Macdonald’s ninth variation to classical group characters of types $A$–$D$ [Gol23]. The paper proves Littlewood-type identities for these generalized characters and obtains Jacobi–Trudi and Nägelsbach–Kostka identities for factorial and rational versions.

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