72 research outputs found
Vogan classes in type
Consider a weighted Coxeter system . Via its associated
Iwahori-Hecke algebra, we may determine the partition of into
Kazhdan-Lusztig cells. In this paper, we use the theory of Vogan classes
introduced by Bonnaf\'e--Geck to obtain a combinatorial description of the left
cells of type when the ratio of the weights of the first to second
generator is . We further give information on the left cells when this
ratio lies in the interval .Comment: 29 pages, significant revision
The Robinson-Schensted Correspondence and -web Bases
We study natural bases for two constructions of the irreducible
representation of the symmetric group corresponding to : the {\em
reduced web} basis associated to Kuperberg's combinatorial description of the
spider category; and the {\em left cell basis} for the left cell construction
of Kazhdan and Lusztig. In the case of , the spider category is the
Temperley-Lieb category; reduced webs correspond to planar matchings, which are
equivalent to left cell bases. This paper compares the images of these bases
under classical maps: the {\em Robinson-Schensted algorithm} between
permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection}
between Young tableaux and reduced webs.
One main result uses Vogan's generalized -invariant to uncover a close
structural relationship between the web basis and the left cell basis.
Intuitively, generalized -invariants refine the data of the inversion set
of a permutation. We define generalized -invariants intrinsically for
Kazhdan-Lusztig left cell basis elements and for webs. We then show that the
generalized -invariant is preserved by these classical maps. Thus, our
result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of
the Robinson-Schensted correspondence.
Despite all of this, our second main result proves that the reduced web and
left cell bases are inequivalent; that is, these bijections are not
-equivariant maps.Comment: 34 pages, 23 figures, minor corrections and revisions in version
Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras
We consider finite W-algebras U(g,e) associated to even multiplicity
nilpotent elements in classical Lie algebras. We give a classification of
finite dimensional irreducible U(g,e)-modules with integral central character
in terms of the highest weight theory for finite W-algebras. As a corollary, we
obtain a parametrization of primitive ideals of U(g) with associated variety
the closure of the adjoint orbit of e and integral central character.Comment: 38 Pages; made some minor correction
The Robinson-Schensted Correspondence and A\u3csub\u3e2\u3c/sub\u3e-web Bases
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n, n, n]: the reduced web basis associated to Kuperberg’s combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n, n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson–Schensted algorithm between permutations and Young tableaux and Khovanov–Kuperberg’s bijection between Young tableaux and reduced webs. One main result uses Vogan’s generalized τ-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized τ-invariants refine the data of the inversion set of a permutation. We define generalized τ-invariants intrinsically for Kazhdan–Lusztig left cell basis elements and for webs. We then show that the generalized τ-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov–Kuperberg’s bijection as an analogue of the Robinson–Schensted correspondence. Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not S3n-equivariant maps
The Robinson-Schensted Correspondence and A2-Web Bases
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n; n; n]: the reduced web basis associated to Kuperberg\u27s combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n; n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson-Schensted algorithm between permutations and Young tableaux and Khovanov-Kuperberg\u27s bijection between Young tableaux and reduced webs.
One main result uses Vogan\u27s generalized T-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized T-invariants refine the data of the inversion set of a permutation. We define generalized T-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized T-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg\u27s bijection as an analogue of the Robinson-Schensted correspondence.
Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not S3n-equivariant maps
Properties of four partial orders on standard Young tableaux
Let SYT_n be the set of all standard Young tableaux with n cells. After
recalling the definitions of four partial orders, the weak, KL, geometric and
chain orders on SYT_n and some of their crucial properties, we prove three main
results: (i)Intervals in any of these four orders essentially describe the
product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. (ii)
The map sending a tableau to its descent set induces a homotopy equivalence of
the proper parts of all of these orders on tableaux with that of the Boolean
algebra 2^{[n-1]}. In particular, the M\"obius function of these orders on
tableaux is (-1)^{n-3}. (iii) For two of the four orders, one can define a more
general order on skew tableaux having fixed inner boundary, and similarly
analyze their homotopy type and M\"obius function.Comment: 24 pages, 3 figure
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