Timeline for Representations of products of symmetric groups
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 12, 2021 at 20:56 | comment | added | John Baez | Right, @TimothyChow. I'm trying to write a paper for category theorists who aren't experts in representation theory, so ideally I could just say what a splitting field is and point to Corollary 41.6 in The Great Tome of Representation Theory, which says "$\mathbb{Q}$ is a splitting field for $S_n$", instead of making them wade through a construction and figure out for themselves that $\mathbb{Q}$ is a splitting field for $S_n$. Luckily Corollary 4.16 of Lorenz's A Tour of Representation Theory says exactly what I want, right after he does the construction. | |
| Apr 12, 2021 at 16:30 | comment | added | Timothy Chow | Various people have essentially said this, but just to state it explicitly: Perhaps the reason you "haven't found a good reference" for "$\mathbb{Q}$ is a splitting field for any symmetric group" is that what the books tend to prove is a much stronger statement: "Here is an explicit combinatorial construction of the all the finite-dimensional irreducible representations of $S_n$ that uses only integers." There are plenty of good references for this latter fact, of which the fact you're interested in is an immediate corollary. | |
| Apr 12, 2021 at 14:17 | answer | added | Benjamin Steinberg | timeline score: 9 | |
| Apr 11, 2021 at 19:47 | answer | added | Andy Putman | timeline score: 6 | |
| Apr 11, 2021 at 19:42 | answer | added | Maxime Ramzi | timeline score: 4 | |
| Apr 11, 2021 at 19:18 | answer | added | Geoff Robinson | timeline score: 9 | |
| Apr 11, 2021 at 19:15 | history | became hot network question | |||
| Apr 11, 2021 at 17:38 | answer | added | Mare | timeline score: 12 | |
| Apr 11, 2021 at 17:32 | answer | added | LSpice | timeline score: 3 | |
| Apr 11, 2021 at 17:27 | comment | added | LSpice | One more: the related question mathoverflow.net/questions/29919/… MO discovered seems on the cusp (haha, says this student of Harish-Chandra's philosopy of cusp forms) of addressing (1). | |
| Apr 11, 2021 at 17:25 | comment | added | LSpice | Also, for (2), every $S_n$-representation already carries a $\mathbb Z$-structure, so that should be even stronger. | |
| Apr 11, 2021 at 17:24 | comment | added | LSpice | Couldn't (1) and (2) be subsumed by showing that the representations of the form $\overline k \otimes_k (\rho_1 \otimes \dotsb \otimes \rho_p)$ give a decomposition of the regular representation, e.g., character-theoretically? (I don't know if this is true, but am optimistic.) | |
| Apr 11, 2021 at 17:18 | history | migrated | from math.stackexchange.com (revisions) | ||
| Apr 10, 2021 at 23:46 | comment | added | hunter | I wonder if (1) and (2) together are enough to imply the desired claim -- conceivably the product could have a non-factorizable representation that becomes isomorphic to a factorizable one over the algebraic closure. (making up the word "factorizable"to mean "satisfying the conclusion of (1)"). | |
| Apr 10, 2021 at 23:41 | answer | added | Ted | timeline score: 2 | |
| Apr 10, 2021 at 23:11 | history | asked | John Baez | CC BY-SA 4.0 |