Schur Elements in Supersymmetrizing Superalgebras

• Schur elements are explicit invariants derived from adapting idempotents, symmetrizers, and supertraces in Z2-graded algebras.
• They enable the construction of supersymmetrizing forms that support both integral and modular representation theory in superalgebra frameworks.
• Applications range from Lie superalgebras to cyclotomic Hecke–Clifford algebras, providing central projections and well-defined modular bases.

Schur elements for supersymmetrizing superalgebras constitute a central concept at the interface of superalgebra representation theory, (super)symmetrizing forms, and the generalization of classical Schur–Weyl duality. They emerge through the adaptation of idempotents, symmetrizers, and supertraces to the context of $\mathbb{Z}_2$-graded algebras, producing explicit invariants and decompositions in the setting of Lie superalgebras and their centralizer algebras, as well as in the structure theory of cyclotomic Hecke–Clifford and related superalgebras. Schur elements for supersymmetrizing superalgebras have direct implications for the construction of symmetrizing and supersymmetrizing forms, the integral and modular representation theory, and the explicit computation of central invariants.

1. Preliminaries: Superalgebras and (Super)Symmetrizing Forms

A superalgebra $\mathcal{A}$ over a commutative ground ring $R$ with $\operatorname{char}(R)\ne 2$ is a $\mathbb{Z}_2$-graded algebra, $\mathcal{A} = \mathcal{A}_0 \oplus \mathcal{A}_1$, with homogeneous elements having parities $|a| \in \{0, 1\}$. The notions of symmetrizing and supersymmetrizing forms distinguish between the even and super (graded) settings:

• A symmetrizing form $t: \mathcal{A} \to R$ satisfies $t(xy)=t(yx)$ for all homogeneous $x, y$. The induced map $\mathcal{A}\to \mathrm{Hom}_R(\mathcal{A}, R)$, $a \mapsto (b\mapsto t(ba))$, is an $(\mathcal{A},\mathcal{A})$-bimodule isomorphism.
• A supersymmetrizing form $t$ satisfies $t(xy) = (-1)^{|x||y|}t(yx)$, and the map $$\mathcal{A} \to \operatorname{Hom}_R(\mathcal{A},R)$$ is an $(\mathcal{A}, \mathcal{A})$-bimodule isomorphism with left action twisted by parity.

In the symmetric setting, the classical Schur element expansion is

$$t=\sum_{V\in Irr(\mathcal{A})} \frac{1}{2^{\delta(V)} s_V}\chi_V$$

where $\delta(V)=0$ or $1$ according to the type ($\mathrm{M}$ or $\mathrm{Q}$) of the simple module $V$. In the supersymmetric case, all simple modules are type $\mathrm{M}$ and

$t = \sum_V \frac{1}{s_V} \suptr_V$

where $\suptr_V$ is the supertrace on $V$ (Li et al., 23 Nov 2025).

2. Schur Symmetrizers for Schur Superalgebras

Schur superalgebras $S(m|n,r)$ realize highest weight categories and their connections to super-analogues of Schur–Weyl duality. For an (m|n)-hook partition $\lambda\vdash r$ and basic $\lambda$-tableau $T$, the symmetrizer $T^{\lambda}[i:j]$ is defined for tableaux $T_i,T_j$ of shape $\lambda$ by

$$T^\lambda[i:j] = \sum_{K\in C(T)} \sum_{P\in R(T)} \mathrm{sgn}(K) X_{i\cdot P,\, j\cdot K}$$

where $i\cdot o = (-1)^{s(i,o)} i o$ incorporates super sign twists and $X_{u,v}$ are dual to basis monomials of the coordinate algebra $A(m|n,r)$. The modified symmetrizers,

$$T^\lambda\{i:j\} = \frac{1}{r(T_i)\,c(T_j)} T^\lambda[i:j]$$

use explicit row and column factorials $r(T_i),\,c(T_j)$ to produce an integral normalization (Marko, 2020).

The $K$-span $A_{\lambda,K}$ of all $T^\lambda[i:j]$ has a $K$-basis indexed by semistandard pairs, and the $\mathbb{Z}$-span of $T^\lambda\{i:j\}$ forms a $\mathbb{Z}$-basis of the integral form $A_{\lambda,\mathbb{Z}}$. These symmetrizers project (generalizing classical Schur elements) onto the summands corresponding to irreducible $S$-supermodules of weight $\lambda$, and are idempotent up to normalization.

3. Schur Elements via Centralizer Algebras and Supersymmetrization

In the framework of classical Lie superalgebras $\mathfrak{g}$, Schur-Weyl duality holds with super-centralizer algebras $A_k$ appropriate to each series:

• $\mathfrak{gl}_{m|n}$: $A_k = \mathbb{C}[S_k]$ (symmetric group)
• $q_n$: $A_k$ = Hecke–Clifford algebra $H_k$
• $\mathfrak{osp}_{2m+1|2n}$: $A_k=B_k(\delta)$ (Brauer algebra)
• $\mathfrak{p}_n$: $A_k = \mathrm{PB}_k(0)$ (periplectic Brauer)

For non-exceptional types, the images of idempotents in $A_k$ under the sequence

$$A_k\xrightarrow{\alpha_k}T^k(\mathfrak{g})^{\mathfrak{g}} \rightarrow Z(U(\mathfrak{g}))$$

yield minimal central projections called Schur elements. In particular:

• For $\mathfrak{gl}_{m|n}$ and $q_n$, these are images of Young and Hecke-Clifford symmetrizers.
• For $\mathfrak{osp}_{2m+1|2n}$, Brauer idempotents produce corresponding central elements.
• For $\mathfrak{p}_n$, central invariants vanish due to the cancellation between pairs (Luo et al., 26 Nov 2024).

The surjectivity of the map from $T(\mathfrak{g})^\mathfrak{g}$ to $Z(U(\mathfrak{g}))$ in each non-exceptional case establishes that all central elements arise from supersymmetrizations of suitable invariants.

4. Schur Elements and Supersymmetrizing Forms in Hecke–Clifford and Sergeev Algebras

The cyclotomic Hecke–Clifford algebra $\mathcal{H}^f_c(n)$ is supersymmetric if $f=f^{(0)}_Q$ (even cyclotomic polynomial), and symmetric under $f=f^{(s)}_Q$ with invertibility conditions. In the semisimple regime, the Schur elements $s_\lambda$ are given by

$s_\lambda = \frac{(-1)^{|\beta|}}{\tau_{r,n}(F_T)}$

where $F_T$ is the seminormal idempotent for the triple $T=(t,\alpha,\beta)$, and $\tau_{r,n}$ is the canonical Frobenius or supersymmetrizing form. Explicit product formulas for $\tau_{r,n}(F_T)$ involve “residue sequences” and Clifford-theoretic data (Li et al., 23 Nov 2025).

A summary of superalgebra types and associated Schur elements is organized in the following table:

Algebra Type	Centralizer $A_k$	Schur Elements Source
$\mathfrak{gl}_{m|n}$	$\mathbb{C}[S_k]$	Young symmetrizers
$q_n$	Hecke–Clifford $H_k$	Hecke–Clifford idempotents
$\mathfrak{osp}_{2m+1|2n}$	Brauer $B_k(\delta)$	Brauer–diagram idempotents
$\mathfrak{p}_n$	Periplectic Brauer $\mathrm{PB}_k(0)$	(Trivial center, no Schur elements)

These formulas extend to degenerate cyclotomic Sergeev algebras, and under Morita superequivalences, give rise to symmetrizing forms on related quiver Hecke algebras of types $A^{(1)}_{e-1}$ and $C^{(1)}_e$.

5. Integral Structures, Modularity, and Filtrations

The integral form $A_{\lambda, \mathbb{Z}}$ in Schur superalgebras, built from the modified symmetrizers $T^\lambda\{i:j\}$, is stable under the action of $S(m|n,r)_{\mathbb{Z}}$ and provides a $\mathbb{Z}$-basis after reduction modulo $p>2$. This modular procedure yields a basis of $A_{\lambda, K}$ over fields $K$ of positive characteristic, with the modified symmetrizers remaining linearly independent and supporting two-sided filtrations indexed by dominant weights. The associated graded pieces are tensor products of simple highest-weight supermodules corresponding to $\lambda$ (Marko, 2020).

6. Explicit Examples and Applications

• For $\mathfrak{gl}_{1|1}$, the center of the enveloping algebra is generated by the Casimir $C_1 = \operatorname{Str}(E) = E_{11}-E_{22}$, corresponding to the Schur element arising from the identity in $S_1$.
• For $\mathfrak{osp}_{3|2}$, even power supertraces $C_{2r} = \operatorname{Str}(F^{2r})$ yield the full center, with the Harish–Chandra isomorphism identifying them with supersymmetric power sums in the eigenvalues on the Cartan.
• In the case of cyclotomic Hecke–Clifford and Sergeev algebras, explicit rational function formulas for the Schur elements provide new symmetrizing forms even in the infinite type settings such as quiver Hecke (KLR) algebras (Li et al., 23 Nov 2025).

These structures generalize classical bideterminants and Gordan–Capelli type straightening operators, providing explicit realizations of supersymmetrizing forms and central projections for a variety of superalgebras and their module categories.

7. Open Problems and Future Directions

Current research focuses on constructing $\mathbb{Z}$-graded integral bases, exploring the connections between supersymmetrizing forms and graded cellularity, and extending the explicit formulas and basis theorems to broader classes, including other types in the Kac classification. The purely even ortho-symplectic series $\mathfrak{osp}_{2m|2n}$ remains exceptional, with only partial central elements obtainable via the described machinery due to the lack of full Brauer–Schur–Weyl duality (Luo et al., 26 Nov 2024). Further applications include the construction of new symmetrizing and supersymmetrizing forms for generalized Hecke algebras and their quiver analogues.