28,786 reputation
356121
bio website math.umass.edu/~jeh
location U. Massachusetts, Amherst
age 74
visits member for 4 years, 5 months
seen 14 hours ago
More-or-less retired professor at UMass Amherst. Basically an algebraist with interests in Lie theory, specifically representation theory and related algebraic geometry.

2d
comment References for crystal bases and Demazure modules in representation theory
This is really a broad question (and needs more tags such as rt.representation-theory and reference-request). There are lots of sources but with different emphases. For instance, Hong-Kang, Introduction to Quantum Groups and Crystal Bases, GSM 42, AMS, 2002. And a relevant book by Kumar, etc.
Jul
19
comment Calculation with weights of $E_6$
@Claudio: In your added paragraph, what do you mean by saying $\Gamma \subset W$? Some points are still a little out of focus, though what you are aiming at is probably OK.
Jul
16
revised Is the Duflo polynomial conjecture open?
edited body; edited tags
Jul
16
comment Is the Duflo polynomial conjecture open?
It would help to give one or two explicit references here. (And I corrected a typo.)
Jul
15
comment Springer Isomorphisms for Adjoint Simple Exceptional Groups
Maybe it works here and even in types $G_2, F_4$, though I'm a bit skeptical about getting this simple algorithm from such arbitrary choices of root ordering. Anyway, aside from type $A_n$ the isogeny class doesn't seem to affect anything; does "exceptional" matter? Note too that there are numerous "Springer isomorphisms", considered by McNinch-Testerman: front.math.ucdavis.edu/0805.2574
Jul
15
comment Unipotent conjugacy classes
The question is a bit unfocused, since "unipotent" is not directly related to Lie group structure (instead it depends on the algebraic group structure, and the count of unipotent classes is then the same in all good characteristics). As Peter Crooks indicates, there is no "simple" formula even in the general or special linear groups.
Jul
14
comment Representations of orthogonal groups vs representations of reflection groups
You may be asking for too much in this broad formulation, since a finite reflection group might embed as a subgroup of many orthogonal groups. Aside from this, it would be helpful to specify the field here, or at least clarify whether you mean "real reflection group" (there being many more "complex reflection groups" to consider).
Jul
12
comment What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?
@Alexander: Though it's not as direct a connection as you might want, there has been subtle work relating classes of a Weyl group $W$ (such as $S_n$) and nilpotent orbits of a corresponding Lie algebra (here parametrized by partitions of $n$). Springer theory then leads to irreducible representations of $W$. The fit is exact for $S_n$. References include R.W. Carter, G.B. Elkington, A note on the parametrization of conjugacy classes, J. Algebra 20 (1972) 350-354; D. Kazhdan, G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988),129-168.
Jul
12
comment Tensor products of two irreducible representations of reductive groups and their inclusions
My last paragraph is accurate, but I was simplifying the original formulation by replacing $\mu^*$ by $\mu$. Anyway, Victor's answer and the fuller version I got from Kumar extract the answers to (1) and (2) mainly from PRV. For (2) it's probably not strictly necessary to interpret the modules as global sections of line bundles, but for this version the best convention is to use $B^-$ rather than $B$. Algebraic group people like Andersen and Jantzen write things this way. Mixing $B, B^-$ gets confusing.
Jul
11
comment Calculation with weights of $E_6$
@Claudio: Can you clarify your notation $A_2$?
Jul
8
comment Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
@Peter: If you send me an email, I can explain in a little more detail the idea coming from Freudenthal's method which seems to be crucial in your observation (but only case-by-case so far): The underlying question is to relate the "support" of a simple root $\alpha_i$ in the set of all $N$ positive roots to the Coxeter number $h=2N/\ell$ when the corresponding fundamental weight $\varpi_i$ is minuscule.
Jul
7
revised Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
added 418 characters in body
Jul
3
revised Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
added 295 characters in body
Jul
3
accepted Number of Richardson orbits in simple Lie algebras of types $E_n$?
Jul
3
comment Number of Richardson orbits in simple Lie algebras of types $E_n$?
Thanks. Your count of Richardson orbits for types $F_4$ and $E_n$ does seem consistent with Hirai. As you say, the literature on induced orbits is more comprehensive, though it still takes some work to see how many distinct Richardson orbits emerge from the larger list of conjugacy classes of Levi subalgebras. It surprises me that the results aren't made more explicit in the literature, but I'll look again at Elashvili's papers.
Jul
2
awarded  Inquisitive
Jul
2
awarded  Curious
Jul
2
revised Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
deleted 1 character in body
Jul
2
answered Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
Jul
1
comment Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
P.S. To be more specific, are you looking at just the fundamental weights corresponding to minuscule representations? (These have just a single orbit of weights under the Weyl group, but don't exist for some types including $E_8$.)