Timeline for Tensor products of Weyl modules in positive characteristic
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2010 at 22:48 | comment | added | George McNinch | For an a priori defn from the point of view of Kostant's $\mathbf{Z}$-form $U_\mathbf{Z}$ of the env. alg of the corresponding complex Lie algebra $\mathfrak{g}_\mathbf{C}$, the Weyl modules arise as the reduction mod $p$ of minimal $U_\mathbf{Z}$-stable lattices in simple modules for $\mathfrak{g}_\mathbf{C}$. | |
| Jun 15, 2010 at 22:48 | comment | added | George McNinch | The modules the OP (presumably) wants to mention are indeed the "dual Weyl mods" AKA "standard mods". These standrd moduless are the representations with arguably the more "natural" defn: they are given by global sections of G-linearized line bundles on G/B. Stndrd modules have simple socle (soc=max'l semisimple sub) while Weyl mods have unique max'l sub. And indeed, the Weyl mods are precisely the contragredients of the stndrd mods (see next comment for a more intrinsic defn of Weyl mods). | |
| Jun 15, 2010 at 22:48 | comment | added | Jim Humphreys | Yes, there are two lines of development, coming from algebraic geometry and from modular representation theory. An important point of contact is the characterization of one type of module as the dual of the other type (with "dual" highest weights involved), which involves an easy application of Kempf's vanishing theorem as found in Jantzen's 1980 Crelle paper. In both geometry and representation theory there are many interesting outgrowths, so it's difficult to credit everyone involved. But another important contributor has been Henning Andersen. | |
| Jun 15, 2010 at 18:28 | comment | added | Torsten Ekedahl | Unless I am mistaken these types of Weyl modules have the irreducible as unique minimal submodule rather than quotient. Consider the case of $S^pV$ which has $V^{(p)}$ as submodule not quotient module. For this reason they seem to be known as dual Weyl modules. | |
| Jun 15, 2010 at 17:35 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |