On bases of irreducible representations of quantum 𝐺𝐿_{𝑛}

Grojnowski, I.; Lusztig, G.

doi:10.1090/conm/139/1197834

Cited by 33 publications

(35 citation statements)

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“…This convention for the action of H r on T is consistent with that in [16,50], but not with that in [22]. Note that v k , T…”

Section: 1mentioning

confidence: 61%

“…In this section we describe the commuting actions of U q (g V ) and H r on T := V ⊗r as in [25,22,50,16] and give several characterizations of the upper canonical basis and projected upper canonical basis of T; we closely follow [16,9] and are consistent with their conventions. This background will be needed to construct an upper canonical basis forΛ rX in §14 and motivates the hypothesized basis forX ⊗r detailed in Conjecture 19.1.…”

Section: Tensor Products Of Based Modulesmentioning

confidence: 99%

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Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

2015

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Abstract. The Kronecker coefficient g λµν is the multiplicity of the GL(V ) × GL(W )-irreducible V λ ⊗ W µ in the restriction of the GL(X)-irreducible X ν via the natural map, where V, W are C-vector spaces and X = V ⊗ W . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients.We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam [1] to construct, in the case dim(V ) = dim(W ) = 2, a representatioň X ν of the nonstandard quantum group that specializes to Res GL(V )×GL(W ) X ν at q = 1. We then define a global crystal basis +HNSTC(ν) ofX ν that solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ, µ) is the Kronecker coefficient g λµν . We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U q (sl 2 ) graphical calculus from [19], and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula in [15]. As a byproduct of the approach, we also obtain a rule for the decomposition of Res GL2×GL2⋊S2 X ν into irreducibles.

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“…This convention for the action of H r on T is consistent with that in [16,50], but not with that in [22]. Note that v k , T…”

Section: 1mentioning

confidence: 61%

Section: Tensor Products Of Based Modulesmentioning

confidence: 99%

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

2015

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“…The role of the canonical basis for the modified quantum gl(N) is similar to that of KazhdanLusztig bases for Iwahori-Hecke algebras. Subsequently, the Schur-Jimbo duality, as a bridge connecting the Iwahori-Hecke algebra of GL(d) and (modified) quantum gl(N), is realized geometrically by considering the product variety of the complete flag variety and the N-step partial flag variety of GL(d) in [GL92]. The modified quantum sl(N) (a variant of quantum gl(N)) and its canonical basis are further categorified in the works [La10] and [KhLa10], which play a fundamental role in higher representation theory and the categorification of knot invariants.…”

Section: Introductionmentioning

confidence: 99%

Geometric Schur duality of classical type, II

Zhang

2015

Trans. Amer. Math. Soc. Ser. B

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Abstract. We establish algebraically and geometrically a duality between the IwahoriHecke algebra of type D and two new quantum algebras arising from the geometry of N -step isotropic flag varieties of type D. This duality is a type D counterpart of the Schur-Jimbo duality of type A and the Schur-like duality of type B/C discovered by Bao-Wang. The new algebras play a role in the type D duality similar to the modified quantum gl(N ) in type A, and the modified coideal subalgebras of quantum gl(N ) in type B/C. We construct canonical bases for these two algebras.

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“…Building upon results in [13] and [25] [18].) Not surprisingly, the dual canonical basis has no entirely elementary description.…”

Section: A Multigrading Of C[x] and Two Basesmentioning

confidence: 97%

Bitableaux and Zero Sets of Dual Canonical Basis Elements

Rhoades

Skandera

2011

Ann. Comb.

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Abstract. We state new results concerning the zero sets of polynomials belonging to the dual canonical basis of C[x 1,1 , . . . , x n,n ]. As an application, we show that this basis is related by a unitriangular transition matrix to the simpler bitableau basis popularized by Désarménien-Kung-Rota. It follows that spaces spanned by certain subsets of the dual canonical basis can be characterized in terms of products of matrix minors, or in terms of their common zero sets.

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On bases of irreducible representations of quantum 𝐺𝐿_{𝑛}

Cited by 33 publications

References 0 publications

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

Geometric Schur duality of classical type, II

Bitableaux and Zero Sets of Dual Canonical Basis Elements

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