Timeline for Representations of general linear groups GL_n(F_q) - decomposition of tensor product?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2017 at 8:30 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
added 174 characters in body
|
| Jun 20, 2017 at 8:23 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
Made title specific to the question. Added tags
|
| Jul 29, 2010 at 14:53 | comment | added | Bruce Westbury | Darij, thanks, that's what I was referring to. | |
| Jul 29, 2010 at 13:05 | comment | added | Jim Humphreys |
@Pooja: For large enough $p$, the "patterns" of decomposition you get depend on the Weyl group but not essentially on $p$ even though the characters involve parameters depending on $p$. Look at GL$(2,p)$ or SL$(2,p)$ for $p$ large: "most" irreducible characters have degree $p-1$ or $p+1$ (roughly half of each), while a typical tensor product decomposes as a sum of roughly $p$ of these in limited patterns. Bookkeeping is tedious even in this simple case, of course, and hopeless in general. Each power$p^m$ expands this list of patterns in a way depending on $m$.
|
|
| Jul 29, 2010 at 13:01 | comment | added | darij grinberg | Bruce: I know almost nothing about these $\mathrm{GL}_n\left(\mathbb F_q\right)$ groups except that they are, in certain ways, similar to $S_n$. But for $S_n$ one can compute interior tensor products through the Hopf algebraic structure of $R\left(S\right)$ inductively by the formula $\left(ab\right)\times c=\left(a\times c_{(1)}\right)\left(b\times c_{(2)}\right)$, where $c_{(1)}\otimes c_{(2)}$ is Sweedler notation for $\Delta c$, and $\times$ denotes the interior tensor product (as opposed to $ab$ which means the exterior tensor product, i. e. the induced one). | |
| Jul 29, 2010 at 12:54 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
edited title
|
| Jul 29, 2010 at 7:59 | comment | added | Pooja Singla | Darij, Jim, and Bruce thanks a lot for your comments. I just realized that representations that I am looking at are infact the ones coming from Shur weyl duality and hence fits into Zelevinsky's set of representations. Hence I got the reply. But still one question is interesting for me for general case: Do the decomposition of $V \times W$ depend on finite field F_q (large enough field)? For example in the above case these do not so was just wondering if this is true in general or not? | |
| Jul 28, 2010 at 20:12 | comment | added | Bruce Westbury | The Zelevinski approach uses parabolic induction to go from $GL_n(F_q)\times GL_m(F_q)$ to $GL_{n+m}(F_q)$. It would take some work to use this to answer the question. The tensor products of representations of symmetric groups are determined by Littlewood-Richardson coefficients but the relationship is complicated. | |
| Jul 28, 2010 at 16:33 | comment | added | Jim Humphreys | P.S. An elegant treatment of Green's ideas is given in Chapter IV of Symmetric Functions and Hall Polynomials by I.G. Macdonald (2nd ed., 1995, Oxford). As in Green's paper, the goal is an efficient description of irreducible characters using lots of combinatorics. But it remains technically challenging to compute an actual character table of any size, or to decompose tensor products, or to carry out branching rules. | |
| Jul 28, 2010 at 16:08 | comment | added | Jim Humphreys | It has to be kept in mind here that the irreducibles themselves are not explicitly "known", but their characters were determined recursively in the classical 1955 Transactions AMS paper of J.A. Green. With this in mind, your problem is to work out the explicit irreducible character decomposition for the product of two explicitly given characters. Zelevinsky's 1981 Springer Lecture Notes No.869 offers an interesting approach to the representations in the spirit of classical Schur-Weyl theory, but without getting into the details of arbitrary tensor product decompositions. | |
| Jul 28, 2010 at 15:13 | comment | added | darij grinberg | Something makes me feel Zelevinsky's "Representations of Finite Classical Groups" is the right source here. | |
| Jul 28, 2010 at 15:04 | history | asked | Pooja Singla | CC BY-SA 2.5 |