Supersymmetric Schur Polynomial

• Supersymmetric Schur polynomial is defined as the generating function of (k,ℓ)-semistandard tableaux with two commuting alphabets subject to hook constraints.
• It satisfies classical symmetric function identities, including determinantal formulas and hook inequalities, that connect tableau combinatorics with polyhedral geometry.
• Its saturated Newton polytope property, achieved via total unimodularity, leads to efficient linear optimization and bridges combinatorics with algebraic geometry.

The supersymmetric Schur polynomial is the standard Schur-type polynomial attached to two alphabets of commuting variables and a partition constrained by a hook condition. In the Berele–Regev framework one fixes $x=(x_1,\dots,x_k)$, $y=(y_1,\dots,y_\ell)$, and a partition $\lambda\in H(k,\ell)$, and defines the polynomial as the generating function of $(k,\ell)$-semistandard tableaux of shape $\lambda$. In the symmetric-function and fermionic literature the same object is commonly written $s_\lambda(x/y)$; in the finite-variable setting, “hook Schur function” is a standard synonym (Hiep et al., 30 Jul 2025, Iwao, 2021, Blasiak et al., 2015).

1. Classical definition and basic identities

In the Berele–Regev setup, the ordered super-alphabet is

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$

and the indexing partition is required to lie in the $(k,\ell)$-hook

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$

A $(k,\ell)$-semistandard tableau of shape $y=(y_1,\dots,y_\ell)$0 is a filling of the Young diagram $y=(y_1,\dots,y_\ell)$1 by letters $y=(y_1,\dots,y_\ell)$2 such that the $y=(y_1,\dots,y_\ell)$3-letters are weakly increasing along rows and strictly increasing down columns, while the $y=(y_1,\dots,y_\ell)$4-letters are weakly increasing down columns and strictly increasing along rows. If

$y=(y_1,\dots,y_\ell)$5

then the supersymmetric Schur polynomial is

$y=(y_1,\dots,y_\ell)$6

with

$y=(y_1,\dots,y_\ell)$7

This tableau formula is the main combinatorial definition in the recent Newton-polytope study (Hiep et al., 30 Jul 2025).

A complementary symmetric-function presentation starts from the complete supersymmetric functions $y=(y_1,\dots,y_\ell)$8, defined by

$y=(y_1,\dots,y_\ell)$9

Within this notation one has the standard identities

$\lambda\in H(k,\ell)$0

$\lambda\in H(k,\ell)$1

and

$\lambda\in H(k,\ell)$2

The same paper also records the splitting formula

$\lambda\in H(k,\ell)$3

which exhibits the behavior of the supersymmetric Schur function under alphabet decomposition (Iwao, 2021).

A determinantal formula due to Moens–Van der Jeugt, recalled in the Newton-polytope paper, is

$\lambda\in H(k,\ell)$4

where

$\lambda\in H(k,\ell)$5

The same source also records the decomposition

$\lambda\in H(k,\ell)$6

although it emphasizes that this decomposition is not the route used for the recent saturated-Newton-polytope theorem (Hiep et al., 30 Jul 2025).

2. Tableau combinatorics, support, and hook inequalities

The tableau expansion gives a direct description of the monomial support. Writing

$\lambda\in H(k,\ell)$7

one sees that a monomial $\lambda\in H(k,\ell)$8 appears if and only if there exists a $\lambda\in H(k,\ell)$9-semistandard tableau of shape $(k,\ell)$0 with content $(k,\ell)$1. Hence

$(k,\ell)$2

The recent support theorem expresses this set purely by linear inequalities. With

$(k,\ell)$3

the hook inequalities are

$(k,\ell)$4

$(k,\ell)$5

together with

$(k,\ell)$6

The point is that these inequalities are not merely necessary. Using Berele–Regev’s mixed Robinson–Schensted correspondence, the paper proves that they are also sufficient, and therefore

$(k,\ell)$7

This “hook description of the support” is the bridge from tableau combinatorics to polyhedral geometry (Hiep et al., 30 Jul 2025).

The proof uses the word

$(k,\ell)$8

inserted by row insertion for $(k,\ell)$9-letters and column insertion for $\lambda$0-letters. The mixed correspondence yields a $\lambda$1-semistandard insertion tableau whose shape $\lambda$2 satisfies

$\lambda$3

Combined with the hook inequalities and $\lambda$4, this forces $\lambda$5 (Hiep et al., 30 Jul 2025).

A classical specialization occurs when $\lambda$6. Then the $\lambda$7-alphabet is empty and

$\lambda$8

while the support reduces to

$\lambda$9

This is the Schur-support description that the paper presents as resembling Rado’s theorem (Hiep et al., 30 Jul 2025).

The worked example

$s_\lambda(x/y)$0

gives

$s_\lambda(x/y)$1

with support

$s_\lambda(x/y)$2

in coordinates $s_\lambda(x/y)$3. The Newton polytope is described there as a regular hexagon in the affine plane $s_\lambda(x/y)$4, with center $s_\lambda(x/y)$5, which already exhibits the absence of lattice-point gaps (Hiep et al., 30 Jul 2025).

3. Newton polytopes and the saturated-Newton-polytope theorem

For a polynomial

$s_\lambda(x/y)$6

its Newton polytope is

$s_\lambda(x/y)$7

and its support is

$s_\lambda(x/y)$8

The polynomial $s_\lambda(x/y)$9 has a saturated Newton polytope if

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$0

Equivalently, the Newton polytope is the integer hull of its support. In the supersymmetric Schur case this is a particularly strong support statement: every lattice point of the convex hull is realized by an actual monomial (Hiep et al., 30 Jul 2025).

The main recent theorem is concise: $t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$1 The proof is entirely polyhedral. Writing $t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$2,

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$3

the support is encoded as

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$4

and the support theorem identifies $t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$5 with $t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$6. One then considers the real polyhedron

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$7

whose inequalities are exactly the hook inequalities, the size equality, and nonnegativity. Thus

$t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$8

The key structural fact is that the constraint matrix $t_1< t_2<\cdots<t_k<u_1<u_2<\cdots<u_\ell,$9 is totally unimodular. The underlying matrix $(k,\ell)$0 is an interval matrix, since each row contains one consecutive block of ones. After a row-sign normalization, $(k,\ell)$1 again becomes an interval matrix, and hence totally unimodular by the Heller–Tompkins / Fulkerson–Gross theorem. The Hoffman–Kruskal criterion then implies that $(k,\ell)$2 is integral: $(k,\ell)$3 Since $(k,\ell)$4, one gets

$(k,\ell)$5

and therefore

$(k,\ell)$6

The paper states that this is, to the authors’ knowledge, the first application of total unimodularity to the SNP problem, and also the first integrality theorem simultaneously addressing two intertwined alphabets in this supersymmetric setting (Hiep et al., 30 Jul 2025).

The same framework has algorithmic consequences. Because $(k,\ell)$7 is integral, every linear optimization problem

$(k,\ell)$8

with $(k,\ell)$9 attains its maximum at an integral vertex $H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$0, hence at the exponent of an actual monomial of $H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$1. The paper also notes that, since the matrix is totally unimodular, such linear programs can be solved in strongly polynomial time using Tardos’ algorithm. It further records a conjectural direction, namely that every supersymmetric Schur function is Lorentzian, and suggests extensions to super-Stanley symmetric functions and supersymmetric Macdonald polynomials (Hiep et al., 30 Jul 2025).

4. Fermionic, determinantal, and integrable realizations

The supersymmetric Schur function admits a standard free-fermionic realization. In the charged fermionic Fock-space formalism with Heisenberg operators $H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$2, one sets

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$3

where

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$4

If

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$5

then

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$6

The conjugation formula

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$7

is the basic mechanism behind this representation (Iwao, 2021).

The same paper embeds supersymmetric Schur functions into the broader family of multi-Schur functions. For row-dependent supersymmetric alphabets

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$8

it defines

$H(k,\ell):=\{\lambda\mid \lambda_{k+1}\le \ell\}.$9

and proves the Jacobi–Trudi-type determinant

$(k,\ell)$0

Ordinary supersymmetric Schur functions are recovered by the specialization

$(k,\ell)$1

so the two-alphabet object appears as the first nontrivial instance of a rowwise supersymmetric theory (Iwao, 2021).

A different generalization is provided by the free-fermionic six-vertex construction of the functions

$(k,\ell)$2

depending on two alphabets and two doubly infinite parameter sequences. Their defining operator formula is

$(k,\ell)$3

and the supersymmetric Schur polynomial is recovered at

$(k,\ell)$4

This framework yields a supersymmetric Cauchy identity,

$(k,\ell)$5

together with the duality

$(k,\ell)$6

The same paper emphasizes that refined Yang–Baxter equations, rather than the classical Yang–Baxter equation alone, are the mechanism behind separate symmetry in the $(k,\ell)$7- and $(k,\ell)$8-variables and the cancellation property characteristic of supersymmetry. In the specialization $(k,\ell)$9, one recovers ordinary supersymmetric Schur identities such as Jacobi–Trudi, Giambelli, the hook generating series, and the Berele–Regev square factorization

$y=(y_1,\dots,y_\ell)$00

(Naprienko, 2023).

5. Skew, factorial, and related generalizations

The skew version already appears in the tableau model with alphabet

$y=(y_1,\dots,y_\ell)$01

where a skew supertableau of shape $y=(y_1,\dots,y_\ell)$02 is weakly increasing along rows and columns, has no repeated unprimed entry in a column, and no repeated primed entry in a row. The ordinary supersymmetric skew Schur function is then

$y=(y_1,\dots,y_\ell)$03

with

$y=(y_1,\dots,y_\ell)$04

This is the starting point for the factorial and “ninth variation” extensions (Foley et al., 2020).

The same paper introduces a generalized ninth variation

$y=(y_1,\dots,y_\ell)$05

where primed and unprimed letters may be interleaved in an arbitrary total order $y=(y_1,\dots,y_\ell)$06, and the weights depend both on the entry and on the content $y=(y_1,\dots,y_\ell)$07. At this level the resulting functions are generally not supersymmetric. Supersymmetry is restored by the factorial specialization

$y=(y_1,\dots,y_\ell)$08

with

$y=(y_1,\dots,y_\ell)$09

Here $y=(y_1,\dots,y_\ell)$10 records a signed imbalance of unprimed and primed letters up to the position $y=(y_1,\dots,y_\ell)$11 in the ordered alphabet (Foley et al., 2020).

The main structural statement is that this factorial skew function is independent of the interleaving order $y=(y_1,\dots,y_\ell)$12, symmetric separately in $y=(y_1,\dots,y_\ell)$13 and in $y=(y_1,\dots,y_\ell)$14, and independent of $y=(y_1,\dots,y_\ell)$15 after the specialization $y=(y_1,\dots,y_\ell)$16, $y=(y_1,\dots,y_\ell)$17. In particular it is genuinely supersymmetric. Setting

$y=(y_1,\dots,y_\ell)$18

recovers the classical supersymmetric skew Schur function $y=(y_1,\dots,y_\ell)$19. The same paper proves a full family of outside-decomposition determinant formulas of Hamel–Goulden type for the ninth variation, so the factorial supersymmetric skew Schur functions inherit Jacobi–Trudi, dual Jacobi–Trudi, and more general determinant identities through this specialization (Foley et al., 2020).

6. Terminological variants and adjacent theories

A persistent source of ambiguity is that closely related names are used for several distinct constructions. The two-alphabet supersymmetric Schur polynomial of Berele–Regev type is only one of them.

Setting	Variables	Labels
Berele–Regev supersymmetric Schur polynomial	Two commuting alphabets $y=(y_1,\dots,y_\ell)$20	Partitions in a $y=(y_1,\dots,y_\ell)$21-hook
Schur functions in superspace / super-Schur polynomials	Commuting and anticommuting variables	Superpartitions or super-Young diagrams
Noncommutative super Schur functions	Barred and unbarred noncommuting letters	Partitions, via Jacobi–Trudi-type formulas

In the noncommutative direction, the paper on Kronecker coefficients defines noncommutative super Schur functions

$y=(y_1,\dots,y_\ell)$22

and develops tableau formulas for them in quotient algebras built from barred and unbarred letters. It does not formally state a specialization theorem to the ordinary commutative supersymmetric Schur polynomial, but it explicitly presents these definitions as patterned on the standard super/supersymmetric elementary, homogeneous, and Schur-type functions, and identifies “hook Schur function” as a standard synonym for the classical commutative object in the finite-variable setting (Blasiak et al., 2015).

A different line of work uses commuting variables together with odd Grassmann variables and superpartitions. One paper introduces two natural Schur-type limits of Macdonald superpolynomials,

$y=(y_1,\dots,y_\ell)$23

and formulates conjectural tableau definitions and a conjectural Pieri rule for them (Blondeau-Fournier et al., 2014). Another constructs four families of Schur functions in superspace recursively by super Bernstein vertex operators, including

$y=(y_1,\dots,y_\ell)$24

thereby extending the classical Bernstein-operator construction of ordinary Schur functions (Alarie-Vézina et al., 2018).

The terminology is extended further in work on supersymmetric polynomial families with odd Grassmann variables. One paper defines super-Schur polynomials $y=(y_1,\dots,y_\ell)$25 by upper-triangularity over a supersymmetric monomial basis and orthogonality for the Schur norm,

$y=(y_1,\dots,y_\ell)$26

with labels given by super-partitions and super-Young diagrams (Galakhov et al., 2024). Another constructs Super-Schur polynomials $y=(y_1,\dots,y_\ell)$27 as common eigenfunctions of cut-and-join operators in the semi-Fock representation of the affine super Yangian $y=(y_1,\dots,y_\ell)$28, characterized by

$y=(y_1,\dots,y_\ell)$29

(Galakhov et al., 2023).

The most reliable distinction, therefore, is between the classical two-alphabet supersymmetric Schur polynomial $y=(y_1,\dots,y_\ell)$30 or $y=(y_1,\dots,y_\ell)$31, and the various superspace or noncommutative “super-Schur” objects. They are historically and structurally related, but they are not identical constructions. The recent Newton-polytope theorem belongs specifically to the Berele–Regev two-alphabet theory (Hiep et al., 30 Jul 2025).