Seminormal forms and cyclotomic quiver Hecke algebras of type A

Abstract

This paper shows that the cyclotomic quiver Hecke algebras of type A, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit “integral” closed formula for the Gram determinants of the Specht modules in terms of the combinatorics associated with the KLR grading. We then use seminormal forms to give a deformation of the KLR algebras of type A. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As an application we construct a new distinguished graded cellular basis of the cyclotomic KLR algebras of type A.

Appendix: Seminormal forms for the linear quiver

In this appendix we show how the results in this paper work when $e=0$ so that $\xi \in K$ is either not a root of unity or $\xi =1$ and K is a field of characteristic zero. In order to define a modular system we have to leave the case where the cyclotomic parameters $Q_1,\ldots ,Q_\ell$ are integral, that is, when $Q_l=[\kappa _l]$ for $1\le l\le \ell$. This causes quite a few notational inconveniences, but otherwise the story is much the same as for the case when $e>0$. We do not develop the full theory of “0-idempotent subrings” here. Rather, we show just one way of proving the results in this paper when $e=0$.

Fix a field K and $0\ne \xi \in K$ of quantum characteristic e. That is, either $\xi =1$ and K is a field of characteristic zero or $\xi ^d\ne 1$ for $d\in \mathbb {Z}$. The multicharge ${\varvec{\kappa }}\in \mathbb {Z}^\ell$ is arbitrary.

Let ${\mathcal {O}}=\mathbb {Z}[x,\xi ]_{(x)}$ be the localisation of $\mathbb {Z}[x,\xi ]$ at the principal ideal generated by x. Let $\fancyscript{K}=\mathbb {Q}(x,\xi )$ be the field of fractions of ${\mathcal {O}}$. Define $\mathcal {H}^\varLambda _n({\mathcal {O}})$ to be the cyclotomic Hecke algebra of type A with Hecke parameter $t=\xi$, a unit in ${\mathcal {O}}$, and cyclotomic parameters

$$\begin{aligned} Q_l=x^l+[\kappa _l],\quad \text {for }1\le l\le \ell , \end{aligned}$$

where, as before, $[k]=[k]_t$ for $k\in \mathbb {Z}$. Then $\mathcal {H}^\varLambda _{n}(\fancyscript{K})=\mathcal {H}^\varLambda _n({\mathcal {O}})\otimes _{\mathcal {O}}\fancyscript{K}$ is split semisimple in view of Ariki’s semisimplicity condition [1]. Moreover, by definition, $\mathcal {H}^\varLambda _{n}(K)\cong \mathcal {H}^\varLambda _n({\mathcal {O}})\otimes _{\mathcal {O}}K$, where we consider K as an ${\mathcal {O}}$-module by setting x act on K as multiplication by zero.

Define a new content function for $\mathcal {H}^\varLambda _n({\mathcal {O}})$ by setting

$$\begin{aligned} C_\gamma =t^{c-r}x^{l}+[\kappa _l+c-r], \end{aligned}$$

for a node $\gamma =(l,r,c)$. We will also need the previous definition of contents below. If ${\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})$ is a tableau and $1\le k\le n$ then set $C_k({\mathfrak t})=C_\gamma$, where $\gamma$ is the unique node such that ${\mathfrak t}(\gamma )=k$.

As in Sect. 2.5, let $\{m_{{\mathfrak s}{\mathfrak t}}\,|\,({\mathfrak s},{\mathfrak t})\in {\mathrm{Std}}^2(\mathcal {P}^{\varLambda }_{n})\}$ be the Murphy basis of $\mathcal {H}^\varLambda _n({\mathcal {O}})$. Then the analogue of Lemma 2.6 is that if $1\le r\le n$ then

$$\begin{aligned} m_{{\mathfrak s}{\mathfrak t}}L_r=C_r({\mathfrak t})m_{{\mathfrak s}{\mathfrak t}}+\sum _{({\mathfrak u},{\mathfrak v})\vartriangleright ({\mathfrak s},{\mathfrak t})}r_{{\mathfrak u}{\mathfrak v}}m_{{\mathfrak u}{\mathfrak v}}, \end{aligned}$$

for some $r_{{\mathfrak u}{\mathfrak v}}\in {\mathcal {O}}$. As in Sect. 3.1 define a $*$-seminormal basis of $\mathcal {H}^\varLambda _{n}(\fancyscript{K})$ to be a basis $\{f_{{\mathfrak s}{\mathfrak t}}\}$ of simultaneous two-sided eigenvectors for $L_1,\ldots ,L_n$ such that $f_{{\mathfrak s}{\mathfrak t}}^{*}=f_{{\mathfrak t}{\mathfrak s}}$.

Define a seminormal coefficient system for $\mathcal {H}^\varLambda _n({\mathcal {O}})$ to be a set of scalars ${\varvec{\alpha }}=\{\alpha _r({\mathfrak s})\}$ satisfying Definition 3.5(a), (b) and such that if ${\mathfrak s}\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})$ and ${\mathfrak u}={\mathfrak s}(r,r+1)\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})$ then

$$\begin{aligned} \alpha _r({\mathfrak s})\alpha _r({\mathfrak u}) = \frac{(1-C_r({\mathfrak s})+tC_r({\mathfrak u}))(1+tC_r({\mathfrak s})-C_r({\mathfrak u}))}{P_r({\mathfrak s})P_r({\mathfrak u})}, \end{aligned}$$

(A1)

where $P_r({\mathfrak s})=C_r({\mathfrak u})-C_r({\mathfrak s})$, and where $\alpha _r({\mathfrak s})=0$ if ${\mathfrak u}\notin {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})$.

As in Theorem 3.9, each seminormal basis of $\mathcal {H}^\varLambda _n(\fancyscript{K})$ is determined by a seminormal coefficient system ${\varvec{\alpha }}=\{\alpha _r({\mathfrak s})\}$, such that

$$\begin{aligned} T_rf_{{\mathfrak s}{\mathfrak t}}=\alpha _r({\mathfrak s})f_{{\mathfrak u}{\mathfrak t}}+\frac{1+(t-1)C_{r+1}({\mathfrak s})}{P_r({\mathfrak s})}f_{{\mathfrak s}{\mathfrak t}}, \quad \text {where }{\mathfrak u}={\mathfrak s}(r,r+1), \end{aligned}$$

together with a set of scalars $\{\gamma _{{\mathfrak t}^{\varvec{\lambda }}}\,|\,{\varvec{\lambda }}\in \mathcal {P}^{\varLambda }_{n}\}$. Notice that $I=\mathbb {Z}$, since $e=0$, so if $\mathbf {i}\in I^n$ then ${\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})$ if and only if $c_r({\mathfrak t})=i_r$ and, in turn, this is equivalent to the constant term of $C_r({\mathfrak t})$ being equal to $[i_r]$, for $1\le r\le n$. Arguing as in Lemma 4.3,

$$\begin{aligned} f_{\mathbf {i}}^{\mathcal {O}}= \sum _{{\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})}\frac{1}{\gamma _{\mathfrak t}}f_{{\mathfrak t}{\mathfrak t}}\in \mathcal {H}^\varLambda _n({\mathcal {O}}). \end{aligned}$$

With these definitions in place all of the arguments in Sect. 4 go through with only minor changes. In particular, if $1\le r\le n$ and $\mathbf {i}\in I^n$ then Definition 4.12 should be replaced by

$$\begin{aligned} \psi ^{\mathcal {O}}_rf_{\mathbf {i}}^{\mathcal {O}}={\left\{ \begin{array}{ll} (T_r+1)\frac{1}{M_r}f_{\mathbf {i}}^{\mathcal {O}},&{}\text {if }i_r=i_{r+1}\\ (T_rL_r-L_rT_r)f_{\mathbf {i}}^{\mathcal {O}},&{}\text {if }i_r=i_{r+1}+1,\\ (T_rL_r-L_rT_r)\frac{1}{M_r}f_{\mathbf {i}}^{\mathcal {O}},&{}\text {otherwise,}\\ \end{array}\right. } \end{aligned}$$

and $y^{{\mathcal {O}}}_rf_{\mathbf {i}}^{\mathcal {O}}=\big (L_r-C_r({\mathfrak t})\big )f_{\mathbf {i}}^{\mathcal {O}}$ where, as before, $M_r=1-L_r+tL_{r+1}$. With these new definitions, if ${\mathfrak s}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})$, for $\mathbf {i}\in I^m$, and $1\le r\le n$ then Lemma 4.19 becomes

$$\begin{aligned} \psi ^{\mathcal {O}}_r f_{{\mathfrak s}{\mathfrak t}}=B_r({\mathfrak s})f_{{\mathfrak s}{\mathfrak t}}+\frac{\delta _{i_ri_{r+1}}}{P_r({\mathfrak s})}f_{{\mathfrak u}{\mathfrak t}}, \end{aligned}$$

where ${\mathfrak u}={\mathfrak s}(r,r+1)$ and

$$\begin{aligned} B_r({\mathfrak s})={\left\{ \begin{array}{ll} \frac{\alpha _r({\mathfrak s})}{1-C_r({\mathfrak s})+tC_{r+1}({\mathfrak s})},&{}\text {if }i_r=i_{r+1},\\ \alpha _r({\mathfrak s})P_r({\mathfrak s}),&{}\text {if }i_r=i_{r+1}+1,\\ \frac{\alpha _r(s)P_r({\mathfrak s})}{1-C_r({\mathfrak s})+tC_{r+1}({\mathfrak s})},&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

Observe that if ${\mathfrak u}={\mathfrak s}(r,r+1)$ is a standard tableau then, using (A1), the definitions imply that

$$\begin{aligned} B_r({\mathfrak s})B_r({\mathfrak u}) = {\left\{ \begin{array}{ll} \frac{1}{P_r({\mathfrak s})P_r({\mathfrak u})},&{}\text {if }i_r=i_{r+1},\\ (1-C_r({\mathfrak s})+tC_r({\mathfrak u}))(1+tC_r({\mathfrak s})-C_r({\mathfrak u})),&{}\text {if }i_r\leftrightarrows i_{r+1},\\ (1+tC_r({\mathfrak s})-C_r({\mathfrak u})),&{}\text {if }i_r\rightarrow i_{r+1},\\ (1-C_r({\mathfrak s})+tC_r({\mathfrak u})),&{}\text {if }i_r\leftarrow i_{r+1},\\ 1,&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

Comparing this with Lemma 4.22, it is now easy to see that analogues of Proposition 4.23 and Proposition 4.24 both hold in this situation. Hence, repeating the arguments of Sect. 4.4, a suitable modification of Theorem A also holds. Similarly, the construction of the bases in Sects. 5 and Sect. 6 now goes though largely without change.