Branching functions of \(A_{n-1}^{(1)}\) and Jantzen-Seitz problem for Ariki-Koike algebras
Let \(v\) be a primitive \(n\)th root of unity, \(0\leq u_0\leq\dots\leq u_{\ell-1}< n\) be \(\ell\) integers and \(v_k= v^{u_k}\) for \(0\leq k<\ell\). Denote \({\mathbf i}= (i_0,\dots, i_{n-1})\) where \(i_k\) is the number of \(u_j\) equal to \(k\). The Ariki-Koike algebra [\textit{S. Ariki} and \textit{K. Koike}, Adv. Math. 106, 216-243 (1994; Zbl 0840.20007)] \({\mathcal H}_m({\mathbf i})= {\mathcal H}_m(v; v_0,v_1,\dots, v_{\ell-1})\) is an analogue of the Iwahori-Hecke algebra for the complex reflection group \(G(\ell,1,m)\). The main result of this paper is the determination of the ireducible representations of \({\mathcal H}_m({\mathbf i})\) which restrict to irreducible representations of the subalgebra \({\mathcal H}_{m-1}({\mathbf i})\). The fact that the irreducible \({\mathcal H}_m({\mathbf i})\)-modules are in one-to-one correspondence with the vertices of the crystal graph \(B(\Lambda)\) of the irreducible \(A_{n-1}^{(1)}\)-module \(V(\Lambda)\) with highest weight \(\Lambda= \sum_{k=0}^{n-1} i_k\Lambda_k\) of level \(\ell\) (where \(\Lambda_k\) denotes the \(k\)th fundamental weight of \(A_{n-1}^{(1)}\)) plays a critical role.
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