On Representations of Algebraic Groups in Characteristic Two

Dowd, Michael F.; Sin, Peter

doi:10.1080/00927879608542648

Cited by 21 publications

(16 citation statements)

References 13 publications

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“…Repeated application of the first claim implies that under the action of L(X) the preimage splits as W ⊕Z, with Z the fixed point space of L(X). Hence Z is X-invariant and affords 0 (10). At this point the proof is completed as in Lemma 5.1.4.…”

Section: Proofmentioning

confidence: 87%

“…We prove the claim by the same argument for each of these, so we give the argument just for case (10) Let R be the maximal submodule of S = Xv , where v is a vector of weight 63. Another application of Lemma 4.2.5 shows that 44 occurs as a composition factor of R. Indeed, Z = Xf α v is a nontrivial image of W X (44).…”

Section: (I)mentioning

confidence: 99%

“…For a given A 1 factor, it follows from Lemma 2.1.6 that 6 = 2 ⊗ 4 does not extend 0. The other modules to consider are of the form 2 ⊗ 4 for Part (vii) follows immediately from [13], and (viii) follows from [10].…”

Section: The Case X = Bmentioning

confidence: 99%

“…Cases (8), (10), and (11) To establish ( * ) Generate with vectors of weight 18 to see that L ↓ X has a submodule M of the form 18 3 /S where S is singular of shape 22/16 f /12 d+e / · · ·. As f ≤ 1 the hypotheses of ( * ) hold which yields the claim.…”

Section: Now Consider Case (3)mentioning

confidence: 99%

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The maximal subgroups of positive dimension in exceptional algebraic groups

Liebeck¹,

Seitz²

2004

Memoirs of the AMS

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Section: Proofmentioning

confidence: 87%

Section: (I)mentioning

confidence: 99%

Section: The Case X = Bmentioning

confidence: 99%

Section: Now Consider Case (3)mentioning

confidence: 99%

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The maximal subgroups of positive dimension in exceptional algebraic groups

Liebeck¹,

Seitz²

2004

Memoirs of the AMS

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“…[99]. The case of 8L (5, k) here, for char k = 2, had previously been treated by Dowd and Sin by computer, along with the other semisimple groups of rank at most 4 in this characteristic [35]. This computational phenomenon, of easier calculations for smaller primes, is one reason why it would be nice to have a result like the original Lusztig conjecture, or, better, the Kato conjecture, that would tell us we do not have to compute for too many primes.…”

Section: ) All Irreducible Representations Of G(q) Over K May Be Obtamentioning

confidence: 96%

Linear and Nonlinear Group Actions, and the Newton Institute Program

Scott

1998

Algebraic Groups and Their Representations

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One of the simplest and most useful notions in mathematics is that of a group action: if G is a group and X is a nonempty set, then an action of G on X (or a G-set structure on X) consists of a multiplication operation G x X -+ X, with the image of a pair (g, x) written as, say, gx, with the following axioms satisfied:(1) 1x = x for all x E X (here 1 EGis the identity element of G)j (2) (gh)x = g(hx) for all g, hE G and all x E X.Equivalently, a group action determines, and is determined by, a homomorphism 9 1-+ [x 1-+ gx] of G into the group of all bijective maps of X onto itself (the 'symmetric' group of all permutations of X). If X has the additional structure of a linear space over a field, we might impose an additional axiom that the mappings [x ~ gx] all be linear. In this case, we say the action of G is linear, or that it defines a linear representation of G, and we call X a module for Gover k. In the linear or nonlinear case, if G and X have some topological structure, we often require that the map G x X ----t X be continuous, or in the algebraic geometry case, a morphism of algebraic varieties or schemes. A main theme of the Newton Institute program! on the representation theory of algebraic and related finite groups, of which this conference volume is a part, may then be stated as follows: even in the simplest case where G is a finite group, and X is a finite set with no linear space or algebraic variety structure, there are, nevertheless, deep connections of these nonlinear actions with linear actions, and of the discrete theory of finite groups and their actions with the theory of algebraic groups and other groups and algebras arising in continuous Lie theory. I hope to explain these connections IThe author would like to thank the NSF for its support and the Newton Institute for its hospitality.

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