Jim Humphreys
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Registered User
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More-or-less retired professor at UMass Amherst.
Basically an algebraist with interests in Lie theory,
specifically representation theory and related
algebraic geometry.
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18h |
accepted | Why are affine Lie algebras called affine? |
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2d |
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Heisenberg Lie algebras Can you clarify the meaning of "complement" here? What happens when you take $M$ to be the 1-dimensional center? |
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2d |
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Source of a formula for tensor product multiplicities? P.S. The paper by King-Wybourne requires the added hypothesis that $V(\lambda)$ (in the above formula I was told about) is self-dual. This is true for the adjoint module in your special case, but I'm uncertain whether the added hypothesis is really needed or not. I wonder whether the newer methods make that clear? |
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May 19 |
revised |
Source of a formula for tensor product multiplicities? added 678 characters in body |
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May 19 |
answered | Why are affine Lie algebras called affine? |
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May 18 |
revised |
Source of a formula for tensor product multiplicities? edited body |
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May 18 |
revised |
Source of a formula for tensor product multiplicities? added 162 characters in body |
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May 18 |
revised |
Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras added 961 characters in body; deleted 4 characters in body; deleted 20 characters in body |
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May 17 |
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Reference request : dimensions of real representations of Lie groups @Samuel: From the context I assume you are interested in finite dimensional representations. These are well-studied, but usually indirectly via their Lie algebras and complexifications. As Dietrich points out, the (relatively easy) dimensions depends on Weyl's formula, for compact or complex Lie groups (or Lie algebras). But working with real forms sometimes doubles dimensions, since irreducible over $\mathbb{R}$ may not mean irreducible over $\mathbb{C}$. There are lots of textbooks, but what works best depends on what you know. |
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May 17 |
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The Jantzen-Schaper theorem @Chris: Jantzen's formalism in Representations of Algebraic Groups, II.5, is classical: use the Weyl group to link non-dominant weight to $\pm$ dominant one. Not similar to his cancellation near walls, etc. The theory connecting Weyl modules and Specht modules is elegant but not easy to compute in large cases (probably nothing is). Anyway, GL and SL behave almost the same, up to characters of the center, so switching groups is mostly bookkeeping. Keep in mind that Jantzen filtrations or analogues involve more than just the knowledge of individual characters. |
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May 17 |
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Generators of $Rep(G)$ @O.: Irreps are parametrized by dominant integral highest weights, formalized in the representation ring (Grothendieck ring). For a simply connected semisimple group (or compact Lie group), it's equivalent to work with the Lie algebra; e.g. Brocker & tom Dieck *Representations of Compact Lie Groups, II.7. or Bourbaki Lie Groups and Lie Algebras, Chap. 8, section 7 (and exercise 27). Fundamental dominant weights yield standard "generators". For a not simply connected group, limit to dominant weights in cosets of the weight lattice mod root lattice. (No easy "generators".) |
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May 17 |
accepted | The Jantzen-Schaper theorem |
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May 16 |
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The Jantzen-Schaper theorem P.S. Jantzen's sum formula is computable but doesn't usually pin down the precise filtration layers. Here it's clear: your Weyl module $V(4,1)$ has Weyl dim 35, with simple quotient on layer 0, of dim 21 (from restricted weights and Steinberg tensor product); layer 1 then has three simples of dim 1, 3, 3, while layer 2 has one simple of dim 7. (Here the five composition factors with multiplicity 1 can most easily be seen from Jantzen's generic decomposition formula, which degenerates preictably near a Weyl chamber wall.) |
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May 16 |
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The Jantzen-Schaper theorem @Chris: $\mathrm{SL}_3$ is standard here. Jantzen's method is subtle: you may get a non-dominant weight by reflecting across a (-$\rho$-shifted) affine hyperplane and must then interpret his $\chi(\mu)$ as $\pm$ linked Weyl character. The sum formula gives formally three terms $\chi(\mu)$, two from reflections relative to highest root with one indirectly linked to the 0 weight (with sign = 1). Draw alcove picture! (Besides RAGS, see his short survey, pp. 291-300 in the 1978 Durham proceedings Finite Simple Groups II, Academic Press, 1980.) |
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May 15 |
answered | The Jantzen-Schaper theorem |
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May 15 |
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The Jantzen-Schaper theorem Note that this is not actually a Ph.D. thesis, but a somewhat lower level Diplom thesis. It has had influence in the literature, which is by now extensive. But I'm doubtful that the original document will add much. It's also unlikely to have been scanned for internet access, but of course there are some paper copies. I believe I still have a copy at UMass. But the later literature is much more likely to be helpful: papers by James-Mathas, Fayer, and others. |
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May 14 |
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weights and exceptional root systems added 474 characters in body |
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May 14 |
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A generalisation of the theorem of Maschke edited tags |
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May 14 |
revised |
weights and exceptional root systems added 182 characters in body |
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May 14 |
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Generators of $Rep(G)$ You can work over any algebraically closed field of characteristic 0 if you want (this ensures complete reducibiity of representations). If $G$ is semisimple and simply connected, the finite dimensional irreducible representations with a fundamental highest weight may suffice for your purpose. Removing the simply connected condition complicates the picture somewhat. Adding a central torus is a further complication, even though its irreducible rational representations are just characters. So it's useful to make your assumptions as precise as possible. |
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May 14 |
accepted | weights and exceptional root systems |
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May 13 |
revised |
weights and exceptional root systems added 6 characters in body; edited body |
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May 13 |
revised |
weights and exceptional root systems added 1017 characters in body; deleted 1 characters in body |
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May 13 |
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weights and exceptional root systems added 316 characters in body; deleted 312 characters in body |
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May 13 |
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weights and exceptional root systems @prochet: I misread the question, so I've edited my answer. |
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May 13 |
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weights and exceptional root systems deleted 57 characters in body; deleted 1 characters in body |
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May 13 |
answered | weights and exceptional root systems |
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May 13 |
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Casselman-Shalika formula for split reductive groups It's always helpful to add a reference to the paper itself, which is freely available at numdam.org: numdam.org/item?id=CM_1980__41_2_207_0 Though I can't point to a helpful further reference, I'd note that in the years since MathSciNet began tracking citations in standard journals, this one has been cited 84 times. |
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May 13 |
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Weyl group of the restriction of scalars of split reductive group added 523 characters in body |
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May 13 |
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a Reference for sylow $p-$subgroup Theorem of GL @unknown: In a research paper, I'd probably just refer to this basic example (it's not a real theorem) as "well known". But if pressed to supply a published reference, I'd still emphasize the elementary nature of the example by citing Exercise 8.9 in J.-P. Serre, Linear Representations of Finite Groups, Springer, 1977 (English translation of an earlier French edition). The computation of group and subgroup orders here is straightforward, as other suggested references indicate. |
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May 11 |
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Source of a formula for tensor product multiplicities? @Allen: Yes, it seems reasonable to get this from the recent methods, but I'm curious about who first wrote this down (and where, why, how). There's a fairly extensive literature, hard to search: Dynkin, Kostant, Dixmier, Joseph, .... Kempf too had papers on tensor product decompositions, from the geometric viewpoint. |
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May 11 |
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a Reference for sylow $p-$subgroup Theorem of GL @unknown: This is far from a research-level question, being well-known for generations and written down in textbooks. All it requires it the easy computation of the group and subgroup orders. Did you try first at math.stackexchange.com? |
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May 9 |
asked | Source of a formula for tensor product multiplicities? |
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May 9 |
accepted | Modular reductions of simple characters |
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May 8 |
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Homomorphisms of Lie groups preserving regularity @Misha: I got distracted by the Lie group language and skipped over the key word "monomorphism". So my comment on centralizers isn't relevant except in the image of the map. |
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May 8 |
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R) @David: A helpful answer, including the clarification about passage to an algebraic group and arithmetic subgroups in the formulation. As Raghunathan's 2004 bibliography shows, the CSP has been solved in the bulk of natural cases where it comes up (especially in his work with Prasad); so it seems to be an overstatement to call the problem "far from solved". Maybe I'm misunderstanding what are the "more general reductive groups" you mention. |
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May 8 |
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Modular reductions of simple characters added 1115 characters in body |
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May 8 |
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Modular reductions of simple characters @Geoff: I guess this is part of the rather general treatment in Feit's section 18? I've never found my way around in this book, though it has a lot of useful material. Maybe not enough concrete examples. |
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May 8 |
revised |
Homomorphisms of Lie groups preserving regularity added 1893 characters in body |
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May 8 |
answered | Homomorphisms of Lie groups preserving regularity |
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May 7 |
answered | Modular reductions of simple characters |
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May 7 |
accepted | Connectedness of Springer Fibers |
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May 7 |
answered | Connectedness of Springer Fibers |
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May 7 |
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quasi-minuscule representations @Marc: This is an efficient short summary (more helpful than what's on Wikipedia too, which by the way has an offbeat reference ignoring Bourbaki's introduction of the term minuscule). To attach the precise weight labels is an easy exercise, given the data on each root system in Bourbaki. In the simply-laced cases the representation is of course just the adjoint representation. |
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May 7 |
accepted | Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras |
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May 7 |
revised |
Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras added 1 characters in body |
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May 7 |
answered | Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras |
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May 7 |
awarded | ● gr.group-theory |
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May 6 |
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group generated by Coxeter elements @Sasha: I don't see a role here for the crystallographic root system, since long and short roots don't enter into the Weyl group structure. Here types $B, C$ coincide. |
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May 6 |
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group generated by Coxeter elements @Sasha: That looks right. I didn't get quite so far in looking at the cases. In a way it's reassuring that the pattern is not totally uniform. I'm also uncertain about the non-Weyl groups $H_3, H_4$. |
