Timeline for Iwahori-Hecke algebras as endomorphism (or convolution) algebra?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2014 at 9:14 | comment | added | user46809 | Not at all. I'm sorry, I didn't express my meaning properly. | |
| Feb 14, 2014 at 7:24 | vote | accept | user46809 | ||
| Feb 14, 2014 at 7:24 | comment | added | user46809 | Yes! this is exactly my question. | |
| Feb 14, 2014 at 7:19 | comment | added | user46809 | Oh, thank you for your reply！ Now the same question as above: In the finite linear group case q=#$\mathbb{F}_q$ (mod p). Is there some special meaning for general $q$? | |
| Feb 13, 2014 at 19:08 | comment | added | Qiaochu Yuan | What does this mean for general $q$? | |
| Feb 13, 2014 at 18:39 | comment | added | Torsten Schoeneberg | To qualify my earlier comment, suppose one has a Tits system $(G, B, N, S)$ such that for each $s \in S$, the double coset $BsB$ is a union of exactly $q$ (mod $p$) left cosets with respect to $B$. Then exercise 24 gives an isomorphism of $H_n(q, k)$ (as special case of the one def'd in exercise 23) to the convolution algebra of $k$-functions on the double cosets $B \backslash G/B$ (def'd in exercise 22). But now the task remains to find such a Tits system. | |
| Feb 13, 2014 at 17:46 | comment | added | Sam Hopkins | @QiaochuYuan: maybe this reference is the one you mean: advmath.pku.edu.cn/EN/article/… | |
| Feb 13, 2014 at 16:30 | comment | added | Marc Palm | Yes, as the right and left B invariant functions on G. look at Piateski-Shapiro... Gl2 over a finite field. Sorry, i didn't read carefully the question. You will be fine if the the characteristic of the coefficient field doesn't divide the cardinality of the group.If u unckeck my answer, I can remove it and put it as a comment. | |
| Feb 12, 2014 at 17:10 | comment | added | Torsten Schoeneberg | Related: mathoverflow.net/q/4547/27465 . Seems that the Bourbaki exercises mentioned in Jean Lecureux' answer (ex. 22--24 to ch. IV §2 of Lie Groups and Lie algebras, I presume) show this in much greater generality ($G$ group with $(B,N)$-pair with Coxeter group $W$ instead of $S_n$, arbitrary coefficients $k$). | |
| Feb 11, 2014 at 17:58 | comment | added | Qiaochu Yuan | IIRC there is a quantum Schur-Weyl duality between a suitable Hecke algebra and a suitable version of the quantum group $U_q(\mathfrak{gl}_n)$; in particular the former should act as endomorphisms of a tensor power of a representation of the latter. I don't know the details though. | |
| Feb 11, 2014 at 8:31 | history | edited | user46809 | CC BY-SA 3.0 |
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| Feb 11, 2014 at 8:25 | history | edited | user46809 | CC BY-SA 3.0 |
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| S Feb 11, 2014 at 5:36 | history | suggested | UwF | CC BY-SA 3.0 |
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| Feb 11, 2014 at 5:35 | review | Suggested edits | |||
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| Feb 11, 2014 at 5:16 | review | First posts | |||
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| Feb 11, 2014 at 4:56 | history | asked | user46809 | CC BY-SA 3.0 |