Cyclotomic Hecke-Clifford Algebra

• Cyclotomic Hecke–Clifford algebras are Z₂-graded algebras defined as cyclotomic quotients of the affine Hecke–Clifford algebra, unifying Hecke and Clifford theory.
• They are constructed with explicit generators and relations, yielding a PBW basis with dimension rⁿ · n! · 2ⁿ.
• Their representation theory underpins symmetric and spin representations, linking cyclotomic structures to categorification and quantum superalgebra modules.

A cyclotomic Hecke–Clifford algebra is a $\mathbb{Z}_2$-graded (super)algebra defined as a cyclotomic quotient of the affine Hecke–Clifford (affine Sergeev) algebra, naturally generalizing both Hecke algebras of complex reflection groups and Clifford–superalgebra structures. The term encompasses both non-degenerate ("quantum") and degenerate ("Sergeev") cases, playing a central role in the representation theory of symmetric and spin-symmetric groups, quiver Hecke superalgebras, and quantum superalgebra categorification. Key combinatorial and homological features include seminormal bases, explicit primitive idempotents, symmetrizing forms, and the orbit method for irreducible module classification (Shi et al., 12 Jan 2025, Li et al., 21 Feb 2025, Li et al., 23 Nov 2025, Savage, 2017).

1. Defining Relations and Structure

Cyclotomic Hecke–Clifford algebras $\mathcal{H}^f_c(n)$ are defined as quotients of the affine Hecke–Clifford algebras by imposing a monic cyclotomic relation on $X_1$, with $r = \deg f$ the level. The base algebra over a field $\Bbbk$ ($\operatorname{char} \Bbbk \neq2$) has generators:

• Even: $T_1,\ldots,T_{n-1}$ ("Hecke") and $X_1^{\pm1},...,X_n^{\pm1}$ ("Cartan" or "Jucys–Murphy").
• Odd: $C_1,\ldots,C_n$ (Clifford).

The relations are:

• Hecke: $T_i^2 = \varepsilon T_i + 1$, $T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1}$, $T_iT_j = T_jT_i$ for $|i - j| > 1$.
• Laurent: $X_iX_j = X_jX_i$, $X_i X_i^{-1} = 1$.
• Clifford: $C_i^2 = 1$, $C_iC_j = -C_jC_i$ for $i\neq j$.
• Hecke–Clifford: Mixed $T_i$, $X_j$, $C_k$ relations incorporating deformation parameter $\epsilon = q-q^{-1}$, as in (Li et al., 21 Feb 2025, Shi et al., 12 Jan 2025, Li et al., 23 Nov 2025).

Cyclotomic relation (for monic $f$ as in (Li et al., 23 Nov 2025)):

$$f(X_1) = \begin{cases} \prod_{i=1}^m (X_1 + X_1^{-1} - \mathtt q(Q_i)), & r = 2m \ (X_1 - 1) \prod_{i=1}^m (X_1 + X_1^{-1} - \mathtt q(Q_i)), & r = 2m + 1 \end{cases}$$

A Poincaré–Birkhoff–Witt (PBW) basis theorem holds: all elements of the form $X^\alpha C^\beta T_w$ ($\alpha_i \in [0,r{-}1]$, $\beta_i\in\mathbb{Z}_2$, $w\in \mathfrak S_n$) form a basis, giving $\dim_\Bbbk \mathcal{H}^f_c(n) = r^n n! 2^n$ (Li et al., 21 Feb 2025, Savage, 2017). The degenerate (Sergeev) case is similarly constructed but with $T_i$ replaced by $s_i$ (simple transpositions), $X_i$ by $x_i$, and $\epsilon=1$ (Shi et al., 12 Jan 2025, Savage, 2017).

2. Clifford Theory and Automorphism Extensions

The cyclotomic Hecke–Clifford algebra generalizes the classical Clifford theory for symmetric and Weyl groups to the setting of Hecke algebras of imprimitive complex reflection groups $G(r,1,n)$ and their "twisted" fixed-point algebras $G(r,p,n)$. Ram–Ramagge’s construction provides an automorphism $\tau$ on $H_{r,1,n}(u_1,...,u_r;q)$, cyclically shifting parameters, with $\tau^p = \mathrm{id}$ under appropriate constraints, leading to the fixed-point subalgebra description (Liu, 2016):

$H_{r,p,n} = H_{r,1,n}^{\langle \tau \rangle}$

Clifford extensions classify simples of $H_{r,p,n}$ in terms of $G$-orbits and stabilizers acting on simple $H_{r,1,n}$-modules (labeled by $r$-multipartitions). This yields a modular description: for multipartition $\lambda$,

• If $e_\lambda$ is the orbit length, restriction of $V(\lambda)$ to $H_{r,p,n}$ splits into $e_\lambda$ inequivalent simples.
• The fixed-point algebra inherits a Clifford-theoretic module structure encoding both Hecke and Clifford symmetry content (Liu, 2016).

3. Representation Theory and Seminormal Bases

In the semisimple regime (deformation parameters satisfying "separation" or admissibility criteria), irreducible supermodules are classified by "cyclotomic multipartitions" determined by the algebra's flavor and parameters (Shi et al., 12 Jan 2025, Li et al., 21 Feb 2025):

• Ordinary multipartition structures index simple modules for $r=2m$, extended to involve strict partitions in "spin" cases ($r=2m+1$ or $r=2m+2$).
• For $\lambda$ a multipartition of $n$, simple modules $D(\lambda)$ admit explicit seminormal forms. Standard tableaux $t\in\Std(\lambda)$ label basis vectors; Clifford and Cartan generators act locally, with $C_j$ flipping parity indices, $X_i$ acting by explicit eigenvalues constructed from residue combinatorics via the $b_-$ quantum-integers.

Construction of seminormal bases and primitive idempotents enables a full matrix-unit description of blocks; for each tableau triple $T=(t, \alpha_t, \beta_t)$, one constructs explicit idempotents $F_T$ as in (Li et al., 21 Feb 2025). This gives diagonally explicit presentations of blocks and central idempotents.

Under the separation condition, the dimension count, via a (super-)Wedderburn argument, implies semisimplicity, with the dimension squared sum of simples matching the full algebra dimension (Shi et al., 12 Jan 2025).

4. Symmetrizing and Supersymmetrizing Forms, Schur Elements

Cyclotomic Hecke–Clifford algebras, in both even and odd level, admit (super)symmetrizing Frobenius forms explicitly constructed via modified "Mackey" traces (Li et al., 23 Nov 2025):

• For $r=2m$, the form $t_{r,n}(h) = \tau_{r,n}(h(X_1\ldots X_n)^m)$ is supersymmetrizing.
• For $r=2m{+}1$ (and invertibility of $(1+X_1)\cdots(1+X_n)$), $$t_{r,n}(h) = \tau_{r,n}(h(X_1\ldots X_n)^m (1+X_1)\cdots(1+X_n))$$ is a symmetrizing form.

In the semisimple case, explicit closed formulas for Schur elements $s_{\underline{\lambda}}$ controlling the behavior of irreducibles with respect to the symmetrizing form are computed in terms of residue data and quantum integers. For $n=1,2$, all such quantities are completely explicit (Li et al., 23 Nov 2025).

The induced forms restrict to Morita-superequivalent cyclotomic quiver Hecke algebras of affine type $A^{(1)}_{e-1}$ and $C^{(1)}_e$, endowing them with explicit symmetric structures.

5. Cyclotomic Quotients as Wreath Product and Quiver Hecke Superalgebras

Cyclotomic Hecke–Clifford algebras are realized as special cases of affine wreath product algebras with Clifford base $F=Cl$, and their cyclotomic quotients correspond to imposing a relation $\prod_{k=1}^r (x_1-u_k)=0$ for a level $r$ parameter set (Savage, 2017):

• The Sergeev algebra, Hecke–Clifford algebra, and quiver Hecke superalgebras (quiver Schur superalgebras) are all unified in this framework.
• PBW and cyclotomic bases, Mackey formula, Frobenius extension properties, and crystal combinatorics for branching are all encompassed in this general theory.
• Classification, functoriality of induction/restriction, and block combinatorics follow from the general affine wreath product structure.

Semisimple cyclotomic Hecke–Clifford algebras possess centers described as symmetric Laurent polynomials in $X_i+X_i^{-1}$ (non-degenerate) or $x_i^2$ (degenerate), admitting classical or spin Fourier theory (Shi et al., 12 Jan 2025, Li et al., 21 Feb 2025).

6. Applications, Special Cases, and Connections

Explicit low-rank computations recover classical spin representation theory and the Clifford-theoretic description of restriction from $B_n$ to $D_n$ in type $B$ Coxeter groups (Liu, 2016). When the cyclotomic level $r=1$, the algebra reduces to the (finite) Sergeev algebra $Cl^{\otimes n} \rtimes S_n$; with the Clifford generators set to zero, one recovers the (degenerate) cyclotomic Hecke algebra (Savage, 2017).

Connections to categorification, crystal theory, and higher representation theory are articulated via the isomorphisms with cyclotomic quiver Hecke superalgebras, with implications for categorified quantum superalgebra modules and Fock space theory (Li et al., 23 Nov 2025).

7. Summary Table: Cyclotomic Hecke–Clifford Algebra Features

Feature	Non-degenerate/Quantum	Degenerate/Sergeev
Generators	$T_i$, $X_i^{\pm1}$, $C_i$	$s_i$, $x_i$, $C_i$
Cyclotomic relation	$f(X_1)$ (monic in $X_1^{\pm1}$)	$g(x_1)$ (monic in $x_1$)
PBW dimension	$r^n n! 2^n$	$r^n n! 2^n$
Simple module labeling	multipartitions/strict partitions	multipartitions/strict partitions
Center	symmetric Laurent polynomials in $X_i+X_i^{-1}$	symmetric polynomials in $x_i^2$
Symmetrizing form	modified Mackey/Frobenius traces	modified Mackey/Frobenius traces
Schur elements	explicit in $\mathrm{res}, b_\pm(\cdot)$	explicit in $\mathrm{res}, b_\pm(\cdot)$

Seminormal forms, explicit matrix units, separation criteria, and complete block decompositions are available in all semisimple cases, and the algebra admits strong functorial and categorification-theoretic properties essential for modern spin and superalgebraic representation theory (Li et al., 21 Feb 2025, Li et al., 23 Nov 2025).