Presenting affine q-Schur algebras

Doty, Stephen; Green, R. M.

doi:10.1007/s00209-006-0076-1

Cited by 29 publications

(67 citation statements)

References 23 publications

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“…Another consequence of the existence of the quotient map (6.3.4) is a simple description of S Q (n; r, s) by generators and relations, generalizing the presentation of S Q (n, d) obtained in [DG2]. We recall the elements ε 1 , .…”

Section: Generators and Relationsmentioning

confidence: 97%

“…In light of results of [DG2] it is now clear how one might define a natural candidate for the rational q-Schur algebra. It is the Q(v)-algebra (v an indeterminate) given by generators E i , F i (1 i n − 1) and…”

Section: Proposition 72 Assume That N 2 and Setmentioning

confidence: 99%

“…In §6 we describe how to obtain cellular bases for rational Schur algebras, by a descent from an ordinary Schur algebra. In §7 we formulate a presentation of S K (n; r, s) by generators and relations (when K has characteristic zero) along the same lines as the earlier result in [DG2] for S K (n, r).…”

Section: Introductionmentioning

confidence: 99%

“…We recall from [DG2,Theorem 1.1] that the algebra S Q (n, d) is isomorphic with the associative algebra with 1 generated by symbols e i , f i (1 i n − 1) and H i (1 i n) subject to the relations (a) …”

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confidence: 99%

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The rational Schur algebra

Dipper

Doty

2008

Represent. Theory

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Abstract. We extend the family of classical Schur algebras in type A, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general linear groups directly through finite dimensional algebras. We show that rational Schur algebras are quasihereditary over any field, and thus have finite global dimension.We obtain explicit cellular bases of a rational Schur algebra by a descent from a certain ordinary Schur algebra. We also obtain a description, by generators and relations, of the rational Schur algebras in characteristic zero.

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Section: Generators and Relationsmentioning

confidence: 97%

Section: Proposition 72 Assume That N 2 and Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The rational Schur algebra

Dipper

Doty

2008

Represent. Theory

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“…In [1] (see also [5]) it is shown that the q-Schur algebra is a homomorphic image θ(U q ) of a modified quantized enveloping algebraU q . ThisU q can be obtained from the quantized enveloping algebra U q of the general linear group, using the triangular decomposition U q = U − U 0 U + of U q , replacing U 0 by a direct sum of infinitely many one-dimensional algebras, generated by idempotents 1 λ , one for each weight λ.…”

Section: Introductionmentioning

confidence: 99%

Codeterminants and q-Schur Algebras

Cliff

Stokke

2008

Algebr Represent Theor

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Abstract. We study codeterminants in the q-Schur algebra S q (n, r) and prove that the standard ones form a basis of S q (n, r), using a quantized version of the Désarménien matrix. We find elements of the form F S 1 λ E T in Lusztig's modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F S ∈ U − and E T ∈ U + are explicitly given products of root vectors, depending on Young tableaux S and T .

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Affine quiver Schur algebras and p-adic $${\textit{GL}}_n$$ GL n

Miemietz¹,

Stroppel²

2019

Sel. Math. New Ser.

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In this paper we consider the (affine) Schur algebra which arises as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of GL2(Q5) in characteristic 3.

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Presenting affine q-Schur algebras

Cited by 29 publications

References 23 publications

The rational Schur algebra

The rational Schur algebra

Codeterminants and q-Schur Algebras

Affine quiver Schur algebras and p-adic $${\textit{GL}}_n$$ GL n

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