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Consider the surjective map of $SL(n,\Bbb Z)$-modules $Hom_{\Bbb C}(V',V)\to Hom_{\Bbb C}(V',V')$. Tim tells us that the identity map from $V'$ to $V'$ lifts to an $f:V'\to V$ which is invariant under …

Let me contribute some confusion. In the situation I am familiar with (algebraic groups), one may first restrict to a Borel subgroup $B$ with the property that $B\cap L$ is a Borel subgroup of $L$. Re …

For the algebraic group $\mathrm{SL}(n)$ over a field of characteristic $p>0$ the (indecomposable) tilting modules are the indecomposable direct summands of tensor products of tensor powers of the fun …

Let me summarize. We take a basis $x$, $y$ of $E$ and the characteristic is $p$. There are two cases where there is a surjective map $E\otimes S^r(E)\to S^{r-1}E$. The first case is when $r=p-1$. The …

There is actually a mildly interesting category in which a given Weyl module is projective. The simplest Weyl module is one dimensional with trivial action. So in that case the category has to have tr …

The result of that book is that the conjecture is true for sufficiently large, but unspecified characteristic. (First fix a Dynkin type.) More recently Peter Fiebig has given actual bounds. See An up …

Indeed the "symmetrized square-free monomials" seem to generate. (Order lexicographically and look what the highest term in a product looks like. Now use that to concoct rewriting rules.) [Oops! T …

For $G=GL(2,\mathbb Z)$ there is no proper subgroup isomorphic to it. Consider the dihedral group $D$ of isometries of a regular 6-gon. There is only one conjugacy class in $G$ of subgroups isomorphic …

> bash-3.2$ LiE > > LiE version 2.2.2 created on Oct 22 2018 at 11:36:00 Authors: Arjeh M. > Cohen, Marc van Leeuwen, Bert Lisser. Purpose: development CWI > > > type '?help' for help information t …

In positive characteristic the only connected groups of finite homological dimension are the tori. We need the following result from Jantzen, Representations of algebraic groups. [J, I 5.13], [J, I …

It is true. The standard reference is the Book by Jantzen, Representations of Algebraic Groups, Second edition. In particular we need the Appendix `Chapter B', and the base change Proposition in part …

The linkage bound is 2. If the algebraic group is simple, say over an algebraically closed $k$, then one has the following lemma. Lemma. If $V$, $W$ are finite dimensional, there is an $m$ dependin …