Representations of finite algebraic groups

Jantzen, Jens Carsten

doi:10.1090/surv/107/08

Cited by 197 publications

(600 citation statements)

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“…We define it using the language of group schemes (see [8]) as a functor from the category of commutative superalgebras to the category of groups: for a commutative superalgebra A let G(A) be the group of all invertible even ( i.e. grading preserving) automorphisms of the A-supermodule V ⊗ A.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

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Crystal structures arising from representations of 𝐺𝐿(𝑚|𝑛)

Kujawa

2006

Represent. Theory

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Abstract. This paper provides results on the modular representation theory of the supergroup GL(m|n). Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category O. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that GL(m|n) has non-conjugate Borel subgroups and we show how Serganova's odd reflections give rise to canonical crystal isomorphisms.

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Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…These functors should be compared with the translation functors defined by Jantzen [8,II.7], Brundan and Kleshchev [3], and Brundan [1]. For ν ∈ P, define O ν p to be the full subcategory of O p of all modules with all their irreducible subquotients isomorphic to L(λ) for some λ ∈ X(T ) with wt(λ) = ν.…”

Section: Theorem 24 Let λ µ ∈ X(t ) Ifmentioning

confidence: 99%

Crystal structures arising from representations of 𝐺𝐿(𝑚|𝑛)

Kujawa

2006

Represent. Theory

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“…Take an algebraic group H over K. Let I be the argumentation ideal of the coordinate ring [20], an element u of hy(H) acts on V by composing The next lemma shows that the surjections (·det) * and (·c 0 ) * between Schur algebra, constructed in the previous subsection, are compatible with ρ r andρ r .…”

Section: Recovering Hyperalgebras From Schur Algebrasmentioning

confidence: 91%

“…We refer to [20] for the definition of hyperalgebra, where it is called the algebra of distributions.…”

Section: Recovering Hyperalgebras From Schur Algebrasmentioning

confidence: 99%

Schur Algebras of Classical Groups II

Liu¹

2010

Communications in Algebra

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We study Schur algebras of classical groups over an algebraically closed field of characteristic not 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C and D, while are not in type B. Consequently Schur algebras of types A, C and D are integral quasi-hereditaryby Donkin [8,10]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras. Finally we calculate some Schur algebras with small parameters and prove Schur-Weyl duality in types B, C and D in a special case.

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“…Let us write w ·μ −μ = ℓλ. Then the image ofμ in X * (T) ⊗ Z R is fixed by the element (ℓλ) · w of the group W ⋉ ℓZΦ, which is the affine Weyl group of [J2,§II.6.1] (for the reductive group which is Langlands dual to G). By a well-known result on groups generated by reflections in real affine spaces (see [Bo, V, §3, Proposition 1]), (ℓλ) · w is a product of reflections in W ⋉ ℓZΦ stabilizingμ.…”

Section: Thenmentioning

confidence: 99%