A generalization of Hall polynomials to ADE case
For a representation \(\pi\) of \(kt\) in a finite-dimensional \(k\)-linear space \(D\), one associates a partition \(\lambda\) with \(\lambda_i=\dim(\text{Im }\pi(t^{i-1}))/\dim(\text{Im }\pi(t^i))\), called the type of \(\pi\). Let \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) be the set of all pairs \((\pi,\underline{D})\) consisting of a representation \(\pi\) of \(kt\) of type \(\lambda\) and \(n\)-step filtration \(\underline{D}=(\{0\}=D^0\subset D^1\subset\cdots\subset D^n=k^{|\lambda|})\) of \(k^{|\lambda|}\) by subrepresentations of \(\pi\) such that \(\pi|_{D^a/D^{a-1}}\) has type \(\mu^a\) for \(a=1,\ldots,n\). If \(k\) is algebraically closed, then \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) is a quasi-projective variety over \(k\). The set \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}({\mathbb F}_{p^l})\) can be identified with the set of \({\mathbb F}_{p^l}\)-rational points of the variety \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(\bar{\mathbb F}_{p^l})\). This is a classical observation of \textit{E. Steinitz} [Zur Theorie der Abelschen Gruppen, Deutsche Math.-Ver. \(9_1\), 80-85 (1901; JFM 32.0149.02)] and \textit{P. Hall} [Proc. 4th Can. Math. Congr. Banff, 1957, 147-159 (1959; Zbl 0122.03403)] that the sets \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}(k)\) have relation to the theory of representations of \(GL(N,{\mathbb C})\) and \(S_N\). In particular, the number of irreducible components of \({\mathfrak N}^{\lambda}_{\mu^1\cdots\mu^n}({\mathbb C})\) is equal to \(c^{\lambda}_{\mu^1\cdots \mu^n}=\dim_{\mathbb C}\Hom_{GL(N,{\mathbb C})} (L_{\lambda},L_{\mu^1}\otimes\cdots\otimes L_{\mu^n})\). NEWLINENEWLINENEWLINEIn the paper under review the author considers other varieties relevant to tensor product multiplicities, for reductive groups other than \(GL(N)\). These varieties were independently described by \textit{H. Nakajima} [Invent. Math. 146, 399-449 (2001; Zbl 1023.17008), see also math.QA/0103008], \textit{M. Varagnolo} and \textit{E. Vasserot} [see Perverse sheaves and quantum Grothendieck rings, Prog. Math. 210. 345-365 (2003); see also math.QA/0103182] and the author [Tensor product varieties and crystals. ADE case, Duke Math. J. 116, 477-524 (2003) see also AG/0103025], respectively. The author describes certain computable polynomials whose leading coefficients are equal to the multiplicities in the tensor product decomposition for representations of Lie algebras of ADE-type.
scientific article; zbMATH DE number 1719041