Cyclotomic Sergeev Algebra

• Cyclotomic Sergeev algebra is a finite-dimensional Z₂-graded algebra defined as a cyclotomic quotient of the degenerate affine Sergeev superalgebra.
• It features natural symmetric and supersymmetric forms that underpin its representation theory, influencing module classification and block theory.
• These algebras are central to spin representation theory and categorification frameworks, linking classical Schur–Weyl duality with modern quiver–Hecke and Heisenberg categorifications.

The cyclotomic Sergeev algebra is a family of finite-dimensional $\mathbb{Z}_2$-graded (super)algebras defined as cyclotomic (i.e., finite, truncated) quotients of the degenerate affine Sergeev (or affine Hecke–Clifford) superalgebras. These algebras play a central role in the spin representation theory of symmetric groups and related Hecke–type algebras, forming the algebraic underpinning for the spin analogs of Schur–Weyl duality, categorifications of Fock spaces, block theory, and connections with quiver–Hecke and Heisenberg categorification frameworks. The structure and representation theory of cyclotomic Sergeev algebras have been developed and unified through works such as (Li et al., 23 Nov 2025, Li et al., 30 Jan 2025), and (Savage, 2017).

1. Definition and Presentation

Let $R$ be a commutative ring with $2$ invertible, $n\geq1$, and let $g(x)\in R[x]$ be a monic degree-$d$ polynomial. Define the affine Sergeev (degenerate Hecke–Clifford) superalgebra $H_n$ as the unital, $\mathbb{Z}_2$-graded $R$-superalgebra generated by:

• Even elements $s_1,\dots,s_{n-1}$ (satisfying the symmetric group relations: $s_i^2=1$, $s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$, $s_is_j=s_js_i$ for $|i-j|>1$)
• Even elements $x_1,\dots,x_n$ (commuting)
• Odd elements $c_1,\dots,c_n$ (Clifford relations: $c_i^2=1$, $c_ic_j=-c_jc_i$ for $i\ne j$)

Mixed relations are:

• $s_i x_i = x_{i+1}s_i - (1 + c_i c_{i+1})$, $s_i x_j = x_j s_i$ for $j\ne i,i+1$
• $s_i c_i = c_{i+1} s_i$, $s_i c_{i+1} = c_i s_i$, $s_i c_j = c_j s_i$ for $j\ne i,i+1$
• $x_i c_i = -c_i x_i$, $x_i c_j = c_j x_i$ for $j \ne i$

The cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is the quotient of $H_n$ by the two-sided ideal generated by $g(x_1)$:

$\mathfrak{h}_n^g = H_n / \langle g(x_1)\rangle.$

This algebra is naturally $\mathbb{Z}_2$-graded: even generators are $x_i$, $s_i$; odd generators are $c_i$ (Li et al., 23 Nov 2025, Li et al., 30 Jan 2025, Savage, 2017).

A PBW-type basis for $\mathfrak{h}_n^g$ is given by monomials in the variables $x_1^{a_1}\cdots x_n^{a_n}c^{\varepsilon}\pi$ with $0\leq a_i< d$, $\varepsilon \in \{0,1\}^n$, and $\pi\in S_n$ (Savage, 2017).

2. Symmetric and Supersymmetric Structures

Cyclotomic Sergeev algebras admit natural symmetrizing structures governed by the parity of $d$:

• If $d$ is odd, there exists an even $R$-linear trace map $t:\mathfrak{h}_n^g\to R$, $t(ab)=t(ba)$, which is a symmetric bilinear form and induces a self-duality between $\mathfrak{h}_n^g$ and its $R$-linear dual as bimodules.
• If $d$ is even, there exists an even $R$-linear form $t$ satisfying $t(ab) = (-1)^{|a||b|} t(ba)$ for homogeneous $a,b$, providing a supersymmetric (Koszul-signed) Frobenius form (Li et al., 30 Jan 2025, Li et al., 23 Nov 2025).

These forms are constructed recursively, using projections (e.g., via PBW bases or Mackey functors), and explicit formulas exist in terms of the algebra's basis elements. In the classical (non-cyclotomic) case, the known Sergeev symmetrizing structures are recovered (Li et al., 23 Nov 2025).

The Nakayama automorphism is trivial when $d$ is even, causing the algebra to be symmetric, and it multiplies Clifford generators $c_i$ by $(-1)^d$ in general (Savage, 2017).

3. Representation Theory and Classification of Simple Modules

Irreducible modules over $\mathfrak{h}_n^g$ are classified, over algebraically closed fields of characteristic $\neq 2$, as follows:

• The simple modules are indexed by multipartitions $$\boldsymbol{\lambda}=(\lambda^{(1)},\dots,\lambda^{(d)})$$ of $n$, with each component a strict partition, reflecting the underlying Clifford action.
• In the semisimple case, the central idempotents are parametrized by these multipartitions, and block theory is controlled by affine crystal data of type $A_{2\ell}^{(2)}$, with blocks labelled by $e$-cores of multipartitions (Savage, 2017, Li et al., 23 Nov 2025).
• Decomposition matrices and modular block theory are controlled via Schur elements (see below); modular reduction of Schur elements determines which irreducibles survive and block membership (Li et al., 23 Nov 2025).

Induction and restriction admit cyclotomic Mackey-type decomposition and the tower of these algebras forms a system of Frobenius extensions, biadjoint up to grading shift (Savage, 2017).

4. Schur Elements and Semisimplicity Criteria

In the semisimple case, explicit formulas for Schur elements $s_\lambda$ with respect to the symmetrizing or supersymmetrizing form $t_{r,n}$ are given:

• The trace functional $t_{r,n}$ decomposes as $$t_{r,n} = \sum_{\lambda}(1/s_\lambda)\cdot \chi_\lambda$$, where $\chi_\lambda$ is the character on the primitive central idempotent labeled by $\lambda$.
• In the nondegenerate ($q$-Hecke–Clifford) case:
  • When the level is even, $$s_\lambda = \prod_{k=1}^n b_{t,k}^{\nu_k m} \cdot [\prod_{\beta}(q(\alpha_k)-q(\beta))] / [\prod_{\alpha\ne\alpha_k}(q(\alpha_k)-q(\alpha))]$$.
  • When the level is odd, $s_\lambda = q(\lambda)^{-1}$ (possibly up to explicit factors of $2$) (Li et al., 23 Nov 2025).
• In the degenerate (Sergeev) case: use $u_\pm=(u\pm u^{-1})/2$ in place of $b$; structural formulas for $s_\lambda$ are provided in terms of $q(\lambda)$, $u_+(\alpha)$, and diagonal components.
• Nonvanishing of all $s_\lambda$ is necessary and sufficient for semisimplicity; block idempotents are sums over multipartitions with identical $p$-residue content.

Schur elements regulate generic degrees, decomposition numbers, graded Cartan matrices, and are central to understanding modular representation theory (Li et al., 23 Nov 2025).

5. Cocenter, Supercocenter, and Center Structure

The cocenter $$\operatorname{Tr}(\mathfrak{h}_n^g) = \mathfrak{h}_n^g / [\mathfrak{h}_n^g, \mathfrak{h}_n^g]$$ admits an explicit basis:

• For both odd and even level, elements are represented by canonical monomials $w_Bc_I$ indexed by colored semi-bipartitions $B$ of $n$ and Clifford compositions.
• For $d$ odd, the cocenter and center coincide (after dualizing), and their rank is given by the number $p_{s,m}(n)$ of suitable strict multipartitions.
• For $d$ even, the supercocenter $\operatorname{SupTr}(\mathfrak{h}_n^g)$ yields a spanning set for the center with upper bound on dimension matching the cardinality of a refined index set $\mathcal{P}_{\mathrm{c}}(d,n)$ (Li et al., 30 Jan 2025).

Ordinary commutator relations are replaced by supercommutators in even level, requiring more refined combinatorics to obtain minimal spanning sets.

Linear independence of the constructed spanning sets is established by reduction to the semisimple generic case (Li et al., 30 Jan 2025).

6. Connections and Applications

Cyclotomic Sergeev algebras generalize and interpolate between several important algebraic structures:

• For $q\to 1$, the nondegenerate cyclotomic Sergeev algebra reduces to the classical Sergeev superalgebra; for vanishing Clifford part, to the Ariki–Koike (cyclotomic Hecke) algebra (Li et al., 23 Nov 2025).
• There exists an isomorphism (Kang–Kashiwara–Tsuchioka) between cyclotomic quiver Hecke–Clifford algebras of type $C^{(1)}_e$ and $\mathfrak{h}_n^g$, and a Morita–super equivalence onto Khovanov–Lauda–Rouquier (KLR) algebras of type $A^{(1)}_{e-1}$.
• The symmetrizing forms constructed on $\mathfrak{h}_n^g$ specialize and generalize the Wan–Wang structure on Hecke–Clifford algebras and the Mathas–Malle form on cyclotomic Hecke algebras.
• These algebras serve as endomorphism algebras in spin Heisenberg categories, controlling higher-level spin Fock space categorifications for types $A^{(2)}$, $B$, and $C$ (Savage, 2017).

These structures enable explicit computation of symmetrizing forms, Cartan invariants, and block theory for spin-type and super analogs of symmetric and Hecke algebras, and they lay the combinatorial foundation for future developments in modular and categorified (super) representation theory.

7. Examples and Low-Rank Computations

Explicit computations in small ranks exhibit the structural features of cyclotomic Sergeev algebras:

Rank $n$	Algebra	Structure/Combinatorics
$n=1$	$\mathrm{Cl}_1\otimes R[x_1]/(g(x_1))$	Direct computation of center, Schur elements
$n=2, m=1$	Level 2 cyclotomic Sergeev	Character table aligns with spin $S_2$
$n=$ arbitrary, $g(x)=x-1$	Hecke–Clifford algebra $H(n)$	Wan–Wang symmetrizing forms, generic degrees

These computations confirm and specialize the general theory, providing concrete illustrations of the parameter and block structure (Li et al., 23 Nov 2025).

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References:

• (Li et al., 23 Nov 2025) On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras
• (Li et al., 30 Jan 2025) On the (super)cocenter of Cyclotomic Sergeev algebras
• (Savage, 2017) Affine wreath product algebras