Defect in cyclotomic Hecke algebras

Abstract

The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated with each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori–Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. We prove that the defect is a block invariant for all cyclotomic Hecke algebras associated with the complex reflection groups of the infinite series G(l, p, n), which include the Weyl groups of type $B_n$ in the unequal parameter case. In particular, for the groups G(l, 1, n), we show that the defect corresponds to the notion of weight in the sense of Fayers. We thus also obtain a new way of computing the weight, which uses a generalisation of the notion of hook lengths. We further show computationally that the defect is a block invariant for all cyclotomic Hecke algebras of exceptional type for which the blocks are known, and we conjecture that the result should hold for all complex reflection groups. Finally, we obtain that the defect is also a block invariant for cyclotomic Yokonuma–Hecke algebras.

Appendix

In Sect. 2.3 we discussed about essential algebras, which are symmetric algebras whose Schur elements are Laurent polynomials of a specific form. Hecke algebras are particular cases of essential algebras. One may ask whether Conjecture 2 holds for other types of essential algebras. The answer is positive for Yokonuma–Hecke algebras (of type A), which were introduced by Yokonuma [71] as generalisations of Iwahori–Hecke algebras.

1.1 The Yokonuma–Hecke algebra of type A and the cyclotomic Yokonuma–Hecke algebra

The Yokonuma–Hecke algebra of type A is a generalisation of the Iwahori–Hecke algebra of type A and can be seen as a particular case of a cyclotomic Yokonuma–Hecke algebra, introduced in [26]. The term “cyclotomic” here does not refer to a cyclotomic specialisation, but to the fact that the cyclotomic Yokonuma–Hecke algebra generalises the Yokonuma–Hecke algebra of type A in the same way that the Ariki–Koike algebra, which is also called “cyclotomic Hecke algebra” by several people, generalises the Hecke algebra of type A. In order not to repeat everything twice, we will work directly with cyclotomic Yokonuma–Hecke algebras, but, for all results used, we will also give the references for the Yokonuma–Hecke algebra of type A, because it was studied first.

Let $n\in \mathbb {Z}_{\ge 0}$ and $d,l \in \mathbb {Z}_{>0}$. Let $\textbf{q}:=(Q_0,\,\ldots ,\,Q_{l-1};\,q)$ be a set of $l+1$ indeterminates and set $R:=\mathbb {Q}[\textbf{q},\textbf{q}^{-1}]$. We define the cyclotomic Yokonuma–Hecke algebra $\textrm{Y}(d,l,n)$ as the associative R-algebra (with unit) with a presentation by generators:

$$\begin{aligned} g_1,g_2,\ldots ,g_{n-1}, t_1,\ldots , t_n, X_1 \end{aligned}$$

and relations:

$$\begin{aligned} \begin{array}{rclcl} g_ig_j &{} = &{} g_jg_i &{}&{} \text{ for } \text{ all } i,j=1,\ldots ,n-1 \text{ such } \text{ that } \vert i-j\vert > 1,\\ g_ig_{i+1}g_i &{} = &{} g_{i+1}g_ig_{i+1} &{}&{} \text{ for } \text{ all } i=1,\ldots ,n-2,\\ t_it_j &{} = &{} t_jt_i &{}&{} \text{ for } \text{ all } i,j=1,\ldots ,n,\\ g_it_j &{} = &{} t_{s_i(j)}g_i &{}&{} \text{ for } \text{ all } i=1,\ldots ,n-1 \text{ and } j=1,\ldots ,n,\\ t_j^d &{} = &{} 1 &{}&{} \text{ for } \text{ all } j=1,\ldots ,n,\\ g_i^2 &{} = &{} q +(q-1) e_{i} g_i &{}&{} \text{ for } \text{ all } i=1,\ldots ,n-1,\\ X_1g_1X_1g_1&{}=&{}g_1X_1g_1X_1&{}&{}\\ X_1g_i&{}=&{}g_iX_1 &{}&{} \text{ for } \text{ all } i=2,\ldots ,n-1,\\ X_1t_j&{}=&{}t_jX_1 &{}&{} \text{ for } \text{ all } j=1,\ldots ,n,\\ (X_1 -Q_0) \cdots (X_1 -Q_{l-1})&{}=&{}0&{}&{} \end{array}\\ \end{aligned}$$

where, for all $i=1,\ldots ,n-1$, $s_i$ is the transposition $(i,i+1)$ and

$$\begin{aligned} e_i=\frac{1}{d}\sum \limits _{0\le s\le d-1}t_i^s t_{i+1}^{-s}. \end{aligned}$$

For $d=1$, the generators $t_j$ disappear and $\textrm{Y}(1,l,n)$ is isomorphic to the Ariki–Koike algebra $R\,\mathcal {H}^\textbf{q}_{n}$. For $l=1$, the generator $X_1$ disappears and $\textrm{Y}(d,1,n)$ is the Yokonuma–Hecke algebra of type A, which in turn becomes the group algebra of G(d, 1, n) for $q=1$. The presentation of $\textrm{Y}(d,1,n)$ by generators and relations is due to [43,44,45], with the simplified quadratic relation for the $g_i$ being due to [25] (see also [24, Remark 3.1] for the quadratic relation given above).

The representation theory of the Yokonuma–Hecke algebra $\textrm{Y}(d,1,n)$ of type A was first studied in [64,65,66], but its irreducible representations were explicitly constructed in [25] and adapted in [24] for the quadratic relation given above. The irreducible representations of $\textrm{Y}(d,l,n)$ were constructed in [26] and can be adapted similarly. It turns out that the algebra $\textrm{Y}(d,l,n)$ is split semisimple over the field $K(\textbf{q})$, where $K:={{\mathbb {Q}}}(\eta _{dl})$ (the splitting field is ${{\mathbb {Q}}}(\eta _{d},\textbf{q})$). Moreover, there exists a bijection $\Pi ^{dl} (n) \leftrightarrow {\textrm{Irr}}(K(\textbf{q})\textrm{Y}(d,l,n)),\, {\varvec{\lambda }}\mapsto V^{{\varvec{\lambda }}}$.

A symmetrising trace $\varvec{\tau }$ was defined in [26] on $K(\textbf{q})\textrm{Y}(d,l,n)$; this map satisfies $\varvec{\tau }(b)=\delta _{1b}$ for all b in a certain basis of $\textrm{Y}(d,l,n)$. The map $\varvec{\tau }$ coincides with the canonical symmetrising trace on $\mathcal {H}^\textbf{q}_{n}$ for $d=1$ and with the symmetrising trace defined on the Yokonuma–Hecke algebra of type A in [25] for $l=1$. By [26, Proposition 7.4], the Schur element of $V^{{\varvec{\lambda }}} \in {\textrm{Irr}}(K(\textbf{q})\textrm{Y}(d,l,n))$ is equal to

$$\begin{aligned} d^n s_{{\varvec{\lambda }}[1]}(\textbf{q}) s_{{\varvec{\lambda }}[2]}(\textbf{q}) \ldots s_{{\varvec{\lambda }}[d]}(\textbf{q}) \end{aligned}$$

(6.1)

where, for all $i=1,\ldots ,d$, ${\varvec{\lambda }}[i]=(\lambda ^{(i-1)l}, \lambda ^{(i-1)l+1}\ldots ,\lambda ^{il-1}) \in \Pi ^l(n)$ and $s_{{\varvec{\lambda }}[i]}(\textbf{q})$ is given by Formula (3.2). Our notation for the ${\varvec{\lambda }}[i]$ is in agreement with the notation used in Sect. 3.3 if we take $\mathcal {U}=\left\{ 0,1,\ldots ,{dl-1}\right\}$, $t=d$ and $\mathcal {U}_i=\{ (i-1)l+k\,|\,k=0,1,\ldots ,l-1\}$ for all $i=1,\ldots ,d$.

This connection between the Schur elements of $\textrm{Y}(d,l,n)$ and those of Ariki–Koike algebras was subsequently explained by the following isomorphism of R-algebras proved independently in [61] and in [62]:

$$\begin{aligned} \textrm{Y}(d,l,n) \cong \bigoplus _{\underset{n_1,\ldots ,n_d\ge 0}{n_1+\ldots +n_d=n}} \textrm{Mat}_{\frac{n!}{n_1!\dots n_d!}} \left( \mathcal {H}^\textbf{q}_{n_1}\otimes _R \mathcal {H}^\textbf{q}_{n_2}\otimes _R \ldots \otimes _R \mathcal {H}_{n_d}^\textbf{q}\right) . \end{aligned}$$

(6.2)

This isomorphism in the case of the Yokonuma–Hecke algebra of type A, that is, the case $l=1$, had first been obtained in [46], subsequently in [41], and adapted in [24] for the quadratic relation given above and the ring R. Another proof for $l=1$ was later given in [29]. The existence of the isomorphism (6.2) implies that the map $\varvec{\tau }$ above is indeed a symmetrising trace on $\textrm{Y}(d,l,n)$ over R.

Let y be an indeterminate and and let $\varphi : K[\textbf{q},\textbf{q}^{-1}] \rightarrow K[y,y^{-1}]$ be a K-algebra morphism such that

$$\begin{aligned}{} & {} \varphi (Q_i)=\eta _{l}^{i}y^{r_{i}} \text { for}\ i=0,1,\ldots ,l-1, \\{} & {} \varphi (q)=y^{r} \end{aligned}$$

where $(r_{0},\ldots ,r_{l-1},r)\in \mathbb {Z}^{l+1}$. Let also $\theta : K[y,y^{-1}] \rightarrow K(\eta ), y \mapsto \eta$ be a specialisation of $K[y,y^{-1}]$ such that $\eta \in {{\mathbb {C}}}^*$ and $\theta (\eta _l)$ is a primitive l-th root of unity (for simplicity, we may assume that $\theta$ is a K-algebra morphism, whence $\theta (\eta _l)=\eta _l$). Let $\Phi$ denote the minimal polynomial of $\eta$ over the field K. It follows from Equations (6.1) and (6.2) that, for ${\varvec{\lambda }}, {\varvec{\mu }}\in \Pi ^{dl}(n)$,

• the $\Phi$-defect of $V^{{\varvec{\lambda }}}$ is equal to $\sum _{i=1}^d \nu _\Phi (\varphi (s_{{\varvec{\lambda }}[i]}(\textbf{q})))$, and

• $V^{{\varvec{\lambda }}}$ and $V^{{\varvec{\mu }}}$ are in the same $\theta$-block if and only $V^{{\varvec{\lambda }}[i]}$ and $V^{{\varvec{\mu }}[i]}$ are in the same $\theta$-block of $K(\eta )(\mathcal {H}^\textbf{q}_{n_i})_\varphi$, where $n_i=|{\varvec{\lambda }}[i]|=|{\varvec{\mu }}[i]|$, for all $i=1,\ldots ,d$.

Given the above, the following result is a direct consequence of Theorem 6.

Theorem 10

Let ${\varvec{\lambda }},{\varvec{\mu }}\in \Pi ^{dl}(n)$. If $V^{{\varvec{\lambda }}}$ and $V^{{\varvec{\mu }}}$ are in the same $\theta$-block, then they have the same $\Phi$-defect.

Therefore, the analogue of Conjecture 2 is also true for the cyclotomic Yokonuma–Hecke algebra $\textrm{Y}(d,l,n)$, and in particular for the Yokonuma–Hecke algebra $\textrm{Y}(d,1,n)$ of type A.

1.2 Other essential algebras

We have stated Conjecture 2 for all cyclotomic Hecke algebras associated with complex reflection groups and we have seen that it holds in all cases for which information on blocks and decomposition matrices is known, that is, for the groups of the infinite series G(l, p, n) and for certain exceptional groups. In the previous subsection we saw that its analogue holds for the Yokonuma–Hecke algebra of type A and for the cyclotomic Yokonuma–Hecke algebra, but one could argue that this is because of the connection, established by the isomorphism (6.2), with the Iwahori–Hecke algebra of type A and the Ariki–Koike algebra respectively. However, one could also wonder whether a similar result holds for all one-parameter essential algebras (including the ones obtained as specialisations of multi-parameter ones as in the end of Sect. 2.3). One could go a step further and wonder whether a similar result holds for any essential algebra, with $\Phi$-defect being replaced by $\Psi _{V,i}(M_{V,i})$-defect, using the notation of Definition 1. Unfortunately, we do not have any data or examples outside the ones treated in this paper to back up this claim. Proving the claim on a generic level would help a lot though in avoiding to do a huge case-by-case analysis in order to prove Conjecture 2 for the exceptional complex reflection groups.

Another perspective to consider is the case of positive characteristic. Throughout this paper we only consider specialisations to subfields of ${{\mathbb {C}}}$. Moreover, one can use classical modular representation theory of finite groupes to see that not all simple modules inside a p-block (where p is a prime number) have the same defect. However, if we consider a cyclotomic Ariki–Koike algebra as in Remark 8 and a specialisation $\theta : {{\mathbb {Z}}}_K[y,y^{-1}] \rightarrow k, y \mapsto \eta$, where k is any field and $\eta \in k$ is a root of unity of order $e>1$, then Theorem 6 is still valid with a proof similar to the one we provided if the characteristic of k is large enough (to be precise, larger than $e+n-1$). It is also valid if $\eta =1$ and k is a field of characteristic $e>0$. These statements can, through the Morita equivalence of Dipper and Mathas, generalise to any cyclotomic Ariki–Koike algebra. We decided not to include these results for the sake of uniformity, and in order to restrict ourselves to the proof of Conjecture 2 in its current form. However, we do not exclude the possibility that other Hecke algebras and, more generally, other essential algebras might share similar properties.