Some remarks on the ring of twisted tilting modules for algebraic groups

Abstract

In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL 2 , Q. J. Math. 56 (2005) 189-207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p > 0. In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.

The ring of twisted tilting modules of quantum $GL_n$

“…As in Section 3, we identify X + (T ) n , X + (T ) ∞ , X + s (T ) n , X + s (T ) ∞ as subsets of X + (T ) × X + ( T ) ∞ in the usual way. By [ [4], p.no 72 (5)] and [ [4] (iii), (iv), (v) and (vi) follows from Proposition of Section 1 of [7].…”

“…We let A C (n) = C ⊗ Z A n . By [7], we know that the polynomials Q p−1 (X i+1 − h(X i ))), i ≥ 0, has no repeated roots. Hence to show that the ring A C (n) is reduced, it is enough to show that (P l−1 (X −1 )(X 0 − g(X −1 )) has no repeated roots.…”

“…If a 0 = 0, then (x 0 − (P l (b) − P l−2 (b)) is a zero divisor in A K (0). Now by [ [7], Section 3, Prop. ], for d = 1 and q = 1 and b = 2, we have (x 0 − 2) is a zero divisor which is not true.…”

“…, where each Ti is an indecomposable tilting module of GL 2 (k). In a paper by [7], it has been shown that the ring of twisted tilting modules of SL 2 (k) is a reduced ring. In [8], it has been proved that the tensor product of any two simple quantum GL 2 (k)-modules can be expressed as a finite direct sum of twisted tilting modules.…”