Invariants for representations of Weyl groups, two-sided cells, and modular representations of Iwahori-Hecke algebras
Let \(\lambda\) be the shape of a Young diagram with corresponding Schur function \(s_\lambda(x)\) in infinite variables, and denote by \(h(v)\) the hook length of \(\lambda\) at \(v\). In [Osaka J. Math. 36, No. 1, 157-176 (1999; Zbl 0915.05008)], \textit{N. Kawanaka} obtained a \(q\)-series identity for \(\sum_\lambda I_\lambda(q)s_\lambda(x)\), where the sum is over all Young diagrams \(\lambda\) and NEWLINE\[NEWLINEI_\lambda(q)=\prod_{v\in\lambda}\tfrac{1+q^{h(v)}}{1-q^{h(v)}}.NEWLINE\]NEWLINE Kawanaka used this identity to find an expression for \(I_\lambda(q)\) in terms of the characters of the symmetric group and the representation of the group by permutation matrices.NEWLINENEWLINENEWLINEDefinition 1.1 of this paper defines a \(q\)-series \(I_W(\chi;q)\) for a Weyl group \(W\) acting on a complex vector space as a reflection group, where \(\chi\) is a character of a finite-dimensional representation \(\pi\) of \(W\). The authors call this the Kawanaka invariant of \(\pi\); it generalizes the aforementioned expression for \(I_\lambda(q)\) to the case of an arbitrary Weyl group.NEWLINENEWLINENEWLINEThe authors establish, in Theorem 2.1, an expression for the Kawanaka invariant in type \(B\) in terms of hook lengths. They conjecture a more complicated closed formula for the Kawanaka invariants in type \(D\) in Conjecture 3.2, and they establish in Theorem 2.2 an expression for these invariants in terms of Littlewood-Richardson coefficients. In Theorem 3.5, a recursive formula for the type \(D\) invariants is given in terms of the combinatorics of partitions. The exceptional and dihedral cases of the invariants have also been explicitly calculated (\S 2.4).NEWLINENEWLINENEWLINEIn \S 4, the authors define an equivalence relation on the characters of the Weyl group (or Hecke algebra). The resulting equivalence classes, which give a more refined partition than that induced by the usual Kazhdan-Lusztig two-sided cells, are called refined two-sided cells. The authors relate these to the Kawanaka invariants and other invariants in Observation 4.8, which is based on the results of computer calculations in previous works of the authors.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00024].
scientific article; zbMATH DE number 1642284