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Feb
19 |
comment |
Problems where we can't make a canonical choice, solved by looking at all choices at once
It should be noted that using the maximal torus $T$ etc. is not necessary to develop Deligne-Lusztig theory. No subsequent expositions of the theory that I know of use these objects, and this indicates that the maximal torus $T$ is not particularly helpful mathematically and seems to be mostly an aesthetic choice. In fact, Deligne and Lusztig quickly make a choice (Section 1.7) and most of the subsequent results are worked out using a choice (which of course doesn't matter). |
Dec
8 |
awarded | Yearling |
Aug
29 |
accepted | Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$? |
Aug
21 |
comment |
Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?
Could you clarify what you mean by: 1) "the leading term" of the comultiplication applied to a monomial (my first guess was $x\otimes1+1\otimes x$, but this doesn't seem to be what you mean); 2) "the leading term does not depend on which monomial we pick"; 3) "each pair of monomials in ... comes from a unique monomial of $f$". Also, where does the minimality of $n$ enter in your argument? It seems that the argument aims to derive a contradiction, but if so, where does the non-smoothness enter (supposedly we want to pick a non-zero nilpotent element at some point)? |
Aug
20 |
asked | Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$? |
Jun
12 |
asked | Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms |
Apr
24 |
awarded | Nice Answer |
Dec
8 |
awarded | Yearling |
Jul
24 |
comment |
Rational structures on the flag variety over a finite field
I do not understand why this question was put on hold as it is perfectly clear what is being asked in the context of common terminology used for algebraic groups over finite fields; see for instance Digne & Michel, Representations of Finite Groups of Lie type, or Malle & Testerman, Linear Algebraic Groups and Finite Groups of Lie Type. |
Jul
2 |
awarded | Curious |
May
7 |
awarded | Fanatic |
Mar
17 |
comment |
Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus
Isn't your remark valid for all reductive $G$ and all $\chi\in X(T)$? |
Feb
3 |
comment |
How can classifying irreducible representations be a “wild” problem?
By the way, doesn't the existence of a good character sheaf theory for $\mathbf{U}(\mathbb{F}_q)$ mean that the irreps are parametrised by objects related to the infinite algebraic group $\mathbf{U}$ in a way which is similar to the case of reductive groups? |
Feb
3 |
comment |
How can classifying irreducible representations be a “wild” problem?
While Lusztig's Jordan decomp of reps of reductive groups over finite fields is important, and the independence of the unipotent reps of $q$ is remarkable, it is not the lack of this which makes a classification problem wild in the technical sense of the original question. E.g., $\mathrm{GL}_2(\mathbb{Z}/p^r)$ has a Jordan decomp of reps (due to Hill) and the 'nilpotent' reps depend on $q$ and even on $r$, but classifying the reps of this group (over any finite local PIR) is a tame problem. |
Feb
2 |
revised |
How can classifying irreducible representations be a “wild” problem?
added 236 characters in body |
Feb
2 |
comment |
How can classifying irreducible representations be a “wild” problem?
I've added some more details about this. |
Feb
2 |
revised |
How can classifying irreducible representations be a “wild” problem?
added 584 characters in body |
Feb
1 |
answered | How can classifying irreducible representations be a “wild” problem? |
Feb
1 |
comment |
How can classifying irreducible representations be a “wild” problem?
I think the difference between the rep. theories of $U_n$ and $\mathrm{GL}_n$ over finite fields is that the irreps. of $U_n(\mathbb{F}_q)$ are inherently more difficult to parametrise (as per Geoff Robinson's comment above). Just knowing the dimensions and multiplicities of the irreps. does not imply that we have a reasonable classification or parametrisation. |
Dec
8 |
awarded | Yearling |