Multiplicities of second cell tilting modules.
Let \(G\) be an almost simple algebraic group over an algebraically closed field \(k\) of prime characteristic \(p\). A tilting module for \(G\) is one which admits a filtration by Weyl modules and a filtration by dual Weyl modules. The indecomposable tilting modules are in one-to-one correspondence with the set of dominant weights, and a fundamental open question is to identify the characters of the indecomposable tilting modules (from which one could compute the characters of simple modules). The main goal of this paper is to obtain a formula for the multiplicity \([Q:T(\lambda)]\) of an indecomposable tilting module \(T(\lambda)\) (corresponding to a dominant weight \(\lambda\)) in an arbitrary tilting module \(Q\). If \(\lambda\) lies in the fundamental alcove, a formula is known which gives this multiplicity in terms of the multiplicity of certain Weyl modules in \(Q\). The approach in this paper is to relate \([Q:T(\lambda)]\) to the analogous multiplicity \([Q_q:T_q(\lambda)]\) for corresponding modules over the associated quantum group. In the quantum setting, \textit{W. Soergel} [Represent. Theory 1, No. 6, 83-114 (1997; Zbl 0886.05123), ibid. 2, No. 13, 432-448 (1998; Zbl 0964.17018)] obtained character formulas for tilting modules. The author makes use of calculations in a module for the Hecke algebra associated to the affine Weyl group introduced by Soergel. The main result here is that \([Q:T(x.\lambda)]=[Q_q:T_q(x.\lambda)]\) for \(\lambda\) in the fundamental alcove and \(x\) in the first or second right cell when the root system is of type \(A_n\) (\(n\geq 2\)), \(D_n\), \(E_n\), and \(G_2\). In the process, some formulas are also given involving wall-crossed tilting modules. Lastly, the author applies the formula in type \(A_n\) to compute the dimension of certain simple \(k\Sigma_r\)-modules by identifying it with the (known) dimension of the corresponding simple module for the Hecke algebra associated to \(\Sigma_r\).
scientific article; zbMATH DE number 2174881
scientific article