The Lusztig Cones of a Quantized Enveloping Algebra of Type A

Marsh, Robert J.

doi:10.1006/jabr.2001.8924

Journal of Algebra

2001

DOI: 10.1006/jabr.2001.8924

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The Lusztig Cones of a Quantized Enveloping Algebra of Type A

Robert J. Marsh

Abstract: We show that for each reduced expression for the longest word in the Weyl group of type A , the corresponding cone arising in Lusztig's description of the n canonical basis in terms of tight monomials is simplicial, and construct explicit spanning vectors. ᮊ

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Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

Marsh

2002

Journal of Algebra

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Let U be the quantum group associated to a Lie algebra g of type A n . The negative part U − of U has a canonical basis B defined by Lusztig and Kashiwara, with favorable properties. We show how the spanning vectors of the cones defined by Lusztig (1993, Israel Math. Conf. Proc. 7, 117-132), when regarded as monomials in Kashiwara's root operators, can be described using a remarkable rectangle combinatorics. We use this to calculate the Lusztig parameters of the corresponding canonical basis elements, conjecturing that translates of these vectors span the simplicial regions of linearity of Lusztig's piecewise-linear function (1990, J. Amer. Math. Soc. 3, 447-498, Sect. 2).  2002 Elsevier Science (USA)

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Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

Marsh

2002

Journal of Algebra

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Canonical basis linearity regions arising from quiver representations

Marsh

Reineke

2003

Journal of Algebra

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In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author [Math. Z. 237 (2001) 639] arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large cone by linear functions arising from these monomial bases for a quantized enveloping algebra.

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Realisation of Lusztig cones

2004

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Let Uq(g) be the quantised enveloping algebra associated to a simple Lie algebra g over C. The negative part U − of Uq(g) possesses a canonical basis B with favourable properties. Lusztig has associated a cone to a reduced expression i for the longest element w 0 in the Weyl group of g, with good properties with respect to monomial elements of B. The first author has associated a subalgebra A i of U − , compatible with the dual basis B * , to each reduced expression i. We show that, after a certain twisting, the string parametrisation of the adapted basis of this subalgebra coincides with the corresponding Lusztig cone. As an application, we give explicit expressions for the generators of the Lusztig cones.

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The Lusztig Cones of a Quantized Enveloping Algebra of Type A

Abstract: We show that for each reduced expression for the longest word in the Weyl group of type A , the corresponding cone arising in Lusztig's description of the n canonical basis in terms of tight monomials is simplicial, and construct explicit spanning vectors. ᮊ

Cited by 3 publications

References 14 publications

Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

Rectangle Diagrams for the Lusztig Cones of Quantized Enveloping Algebras of Type A

Canonical basis linearity regions arising from quiver representations

Realisation of Lusztig cones

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