---
title: Cyclotomic Hecke-Clifford Algebra
url: https://www.emergentmind.com/topics/cyclotomic-hecke-clifford-algebra
type: topic
---

# Cyclotomic Hecke-Clifford Algebra

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 [2501.06763, 2502.15170, 2511.18395, 1709.02998].

## 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 [2502.15170, 2501.06763, 2511.18395].

Cyclotomic relation (for monic $f$ as in [2511.18395]):

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

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 $\dim_\Bbbk \mathcal{H}^f_c(n) = r^n n! 2^n$ [2502.15170, 1709.02998]. The degenerate (Sergeev) case is similarly constructed but with $T_i$ replaced by $s_i$ (simple transpositions), $X_i$ by $x_i$, and $\epsilon=1$ [2501.06763, 1709.02998].

## 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 [1606.05411]:

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

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 [1606.05411].

## 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 [2501.06763, 2502.15170]:
- 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 [2502.15170]. 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 [2501.06763].

## 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 [2511.18395]:
- 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 [2511.18395].

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 [1709.02998]:
- 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 [2501.06763, 2502.15170].

## 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 [1606.05411]. 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 [1709.02998].

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 [2511.18395].

## 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 [2502.15170, 2511.18395].

Source: https://www.emergentmind.com/topics/cyclotomic-hecke-clifford-algebra