Stable centres of Iwahori-Hecke algebras of type A (Q6187529)

In representation theory of finite groups it is important to look at the centre \(Z(RG)\) of the group algebra \(RG\) of a finite group \(G\) over a commutative ring \(R\). As we know the set of all the conjugacy class sums \(\widehat C\)s of \(G\) becomes an \(R\)-basis of \(Z(RG)\) where \(C\) is a conjugacy class of \(G\) and \(\widehat C:=\sum_{g\in C}\,g\in RG\). Throughout \(n\) denotes a positive integer with \(n\geq 2\). In the paper under review the author looks at, first of all, the centre \(Z(\mathbb ZS_n)\) of the group algebra \(\mathbb ZS_n\) of the symmetric group \(S_n\) of degree \(n\) over \(\mathbb Z\), and then at the centre \(Z(H_n(q))\) of the Iwahori-Hecke algebra \(H_n(q)\) of type \(A_{n-1}\). The \(\mathbb Z[q,q^{-1}]\)-algebra \(H_n(q)\) is a \(q\)-analogue of \(\mathbb ZS_n\) because \(H_n(1)\cong\mathbb ZS_n\). Now \textit{M. Geck} and \textit{R. Rouquier} [Prog. Math. 141, 251--272 (1997; Zbl 0868.20013)] construct a \(q\)-analogue of conjugacy class sums in \(\mathbb ZS_n\) for \(H_n(q)\), and the set of these elements becomes a \(\mathbb Z[q,q^{-1}]\)-basis of \(Z(H_n(q))\), that is called the Geck-Rouquier basis of \(Z(H_n(q))\). \textit{H. K. Farahat} and \textit{G. Higman} [Proc. R. Soc. Lond., Ser. A 250, 212--221 (1959; Zbl 0084.03004)] define the Farahat-Higman algebra FH whose \(\mathbb Z\)-algebra epimorphic image is \(Z(\mathbb ZS_n)\). By using this, the algebra FH is \(q\)-deformed by \textit{P.-L. Méliot} [in: Proceedings of the 22nd annual international conference on formal power series and algebraic combinatorics, FPSAC 2010, San Francisco, USA, August 2--6, 2010. Nancy: The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS). 921--932 (2010; Zbl 1374.05241)], that is denoted by FH\(_q\). Just as for FH and \(\mathbb ZS_n\), \(Z(H_n(q))\) is an epimorphic image of FH\(_q\) as \(\mathbb Z[q,q^{-1}]\)-algebras. On the other hand, the author has already given an explicit isomorphism between the \(\mathcal R\)-algebras FH and \(\mathcal R\otimes_{\mathbb Z}\Lambda\), where \(\mathcal R\) is the ring of integer-valued polynomials (that is, it is the subring of \(\mathbb Q[t]\) consisting all polynomials \(p(t)\) satisfying \(p(m)\in\mathbb Z\) for every \(m\in\mathbb Z\)) and \(\Lambda\) is the ring of symmetric functions. The main result of this paper under review is that there is a \(q\)-analogue of the above isomorphism, namely there is an explicit \(\mathcal R[q,q^{-1}]\)-algebra isomorphism between FH\(_q\) and \(\mathcal R[q,q^{-1}] \otimes_{\mathbb Z}\Lambda\), where \(\mathcal R[q,q^{-1}]:=\mathcal R\otimes_{\mathbb Z}\mathbb Z[q,q^{-1}]\). Three small comments: At line \(-8\) on page 2354, `if and only' should be `if and only if'. The reviewer wonders why the term `stable centre' does not show up at all except the title (`stable centre' reminds the reviewer of the term defined by \textit{M. Broué} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 1--26 (1994; Zbl 0827.20007)]). In the references of 12, 13 and 15 the first and family names of the authors are swapped in a wrong way.