Tensor product varieties and crystals: GL case
Let \(\lambda\) be a partition of an integer \(k\) into \(N\) parts, and let \(\mu^1,\mu^2,\dots,\mu^l\) be partitions of integers \(k^1,k^2,\dots,k^l\) into \(N\) parts each, where \(k^1+k^2+\cdots+k^l=k\). Let \(S_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\) be the variety of all \(l\)-step partial flags in \(\mathbb{C}^k\) with dimensions of the subfactors given by \(k^1,k^2,\dots,k^l\) and such that \(t\) preserves each subspace in the flag, and when restricted to the subfactors it defines operators with the Jordan forms given by \(\mu^1,\mu^2,\dots,\mu^l\). This variety is known as the Spaltenstein variety; Spaltenstein proved that it has pure dimension.NEWLINENEWLINENEWLINELet \(C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\) be the set of irreducible components of \(S_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)\). For any partition \(\nu\) of \(k\) into \(N\) parts, let \(L_\nu\) denote the simple \(\text{GL}_N(\mathbb{C})\) representation over \(\mathbb{C}\) with highest weight \(\nu\), and let \(\rho_\nu\) denote the simple \(\mathbb{C} S_k\)-module corresponding to \(\nu\). It follows from \textit{P. Hall} [Proc. 4th Can. Math. Congr. Banff 1957, 147-159 (1959; Zbl 0122.03403)] that: NEWLINE\[NEWLINE|C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)|=\dim_\mathbb{C}\Hom_{\text{GL}_N(\mathbb{C})}(L_{\mu^1}\otimes\cdots\otimes L_{\mu^l},L_\lambda),NEWLINE\]NEWLINE and that NEWLINE\[NEWLINE|C_l((\mu^1,\mu^2,\dots,\mu^l),\lambda)|=\dim_\mathbb{C}\Hom_{S_{k^1}\times\cdots\times S_{k^l}}(\rho_{\mu^1}\otimes\cdots\otimes\rho_{\mu^l},\text{res}^{S_{k^1}\times\cdots\times S_{k^l}}_{S_k}\rho_\lambda).NEWLINE\]NEWLINE Schur-Weyl duality implies the equality of the right hand sides, and either equality can be derived combinatorially, but it is interesting to try to interpret these two formulas geometrically, and this is the aim of the author of the paper under review. Some results in this direction have already been achieved -- see, for example, \textit{W. Borho} and \textit{R. Macpherson} [Astérisque, 101-102, 23-74 (1983; Zbl 0576.14046)].NEWLINENEWLINENEWLINELet \(t\) be a nilpotent operator on \(\mathbb{C}^k\) with nilpotent class \(\lambda\), and let \(M_N(\lambda)\) denote the variety of all \(N\)-step partial flags in \(\mathbb{C}^k\) such that \(t\) preserves all of the subspaces in the flag. Kashiwara and Saito have defined a \({\mathfrak{gl}}_N\)-crystal structure on the set \(\mathbb{M}_N(\lambda)\) of irreducible components of \(M_N(\lambda)\) which is isomorphic to the canonical basis crystal of \(L(\lambda)\).NEWLINENEWLINENEWLINEThe author's approach is to show that the first isomorphism above follows from a geometric crystal theory. In particular, he constructs a variety \(I_N(\mu^1,\dots,\mu^l)\) such that the set \(T_N(\mu^1,\dots,\mu^l)\) of its irreducible components can be equipped with a structure of \({\mathfrak{gl}}_N\)-crystal so that there are the following two isomorphisms: NEWLINE\[NEWLINET_N(\mu^1,\dots,\mu^l)\cong\mathbb{M}_N(\mu^1)\otimes\cdots\otimes\mathbb{M}_N(\mu^l),NEWLINE\]NEWLINE and NEWLINE\[NEWLINET_N(\mu^1,\dots,\mu^l)\cong\bigoplus_\lambda C_l((\mu^1,\dots,\mu^l),\lambda)\otimes\mathbb{M}_N(\lambda),NEWLINE\]NEWLINE where the set \(C_l(\mu^1,\dots,\mu^l)\) is considered as a trivial crystal. These two crystal isomorphisms give a geometric crystal realisation of the first isomorphism above. This geometric understanding gives a possible route to generalisations to Kac-Moody algebras other than \({\mathfrak{gl}}_N\).
scientific article; zbMATH DE number 1680730
scientific article