98 research outputs found
Rigged configurations of type and the filling map
No full textInternational audienceWe give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals in type by using virtualization into type . We consider a special case of this bijection with , and we obtain the so-called Kirillov–Reshetikhin tableaux model for the Kirillov–Reshetikhin crystal.Nous donnons une bijection prservant les statistiques entre les configurations gréées et les produits tensoriels de cristaux de Kirillov–Reshetikhin de type , via une virtualisation en type . Nous considérons un cas particulier de cette bijection pour et obtenons ainsi les modèles de tableaux appelés Kirillov–Reshetikhin pour le cristal Kirillov–Reshetikhin
Crystals for shifted key polynomials
No full textThis article continues our study of P- and Q-key polynomials, which are (non-symmetric) “partial” Schur P- and Q-functions as well as “shifted” versions of key polynomials. Our main results provide a crystal interpretation of P- and Q-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra qn. In the P-key case, the ambient normal crystals are the qn-crystals studied by Grantcharov et al., while in the Q-key case, these are replaced by the extended qn-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into P- and Q-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal qn-crystals and Demazure gln-crystals.</p
Crystals for shifted key polynomials
No full textThis article continues our study of - and -key polynomials, which are
(non-symmetric) "partial" Schur - and -functions as well as "shifted"
versions of key polynomials. Our main results provide a crystal interpretation
of - and -key polynomials, namely, as the characters of certain connected
subcrystals of normal crystals associated to the queer Lie superalgebra
. In the -key case, the ambient normal crystals are the
-crystals studied by Grantcharov et al., while in the -key
case, these are replaced by the extended -crystals recently
introduced by the first author and Tong. Using these constructions, we propose
a crystal-theoretic lift of several conjectures about the decomposition of
involution Schubert polynomials into - and -key polynomials. We verify
these generalized conjectures in a few special cases. Along the way, we
establish some miscellaneous results about normal -crystals and
Demazure -crystals.Comment: 60 pages, 6 figure
Colored lattice models for Demazure characters in types ABC (Recent developments in Combinatorial Representation Theory)
No full textCharacters of irreducible representations of ₙ, the Schur functions are wellknown to be described as partition functions of solvable lattice models. One method to prove this uses the Gelfand-Tsetlin interpretation of semistandard tableaux, which are naturally in bijection with states of the lattice model. This approach has been extended to type C in 2012 by Ivanov using Proctor patterns. A recent trend in solvable lattice models has been to use "colored" lattice models that are based on the R-matrices of finite dimensional quantum affine general linear group (ₙ) representations instead of (₂). In this note, we obtain Demazure characters, which are representations of the corresponding Borel subalgebra and can be thought of as "partial" versions of the irreducible simple Lie algebra representations, and their Demazure atoms analogs by using a colored lattice model built from that of Bump-Brubaker-Buciumas-Gustafsson for type A Demazure atoms. A key component of these lattice models is a natural relationship with the wiring diagram of the defining element of the corresponding Weyl group. We then will discuss the colored version of Ivanov's lattice model to obtain Demazure characters in type B and C and the related tableaux
Cellular subalgebras of the partition algebra
No full textWe describe various diagram algebras and their representation theory using
cellular algebras of Graham and Lehrer and the decomposition into half
diagrams. In particular, we show the diagram algebras surveyed here are all
cellular algebras and parameterize their cell modules. We give a new
construction to build new cellular algebras from a general cellular algebra and
subalgebras of the rook Brauer algebra that we call the cellular wreath
product.Comment: 43 pages, 4 tables; v2, added additional references, expanded proof
of Theorem 3.2, other improvement
Uniform description of the rigged configuration bijection
No full textWe give a uniform description of the bijection \Phi from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form \bigotimes _{i=1}^N B^{r_i,1} in dual untwisted types: simply-laced types and types A_{2n-1}^{(2)}, D_{n+1}^{(2)}, E_6^{(2)}, and D_4^{(3)}. We give a uniform proof that \Phi is a bijection and preserves statistics. We describe \Phi uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that \Phi is a bijection for \bigotimes _{i=1}^N B^{r_i,s_i} when r_i, for all i, map to 0 under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov–Reshetikhin crystals B^{r,1} using tableaux of a fixed height k_r depending on r in all affine types. Additionally, we are able to describe crystals B^{r,s} using k_r \times s shaped tableaux that are conjecturally the crystal basis for Kirillov–Reshetikhin modules for various nodes r
A crystal to rigged configuration bijection and the filling map for type D-4(3)
No full textWe give a bijection Phi from rigged configurations to a tensor product of Kirillov Reshetikhin crystals of the form B-r,B-1 and B-1,B-s in type D-4((3)). We show that the cocharge statistic is sent to the energy statistic for tensor products circle times(N)(i=1) B-r,B-1 and circle times(N)(i=1) B-1,B-s. We extend this bijection to a single B-r,B-s, show that it preserves statistics, and obtain the so-called Kirillov-Reshetikhin tableaux model for B-r,B-s. Additionally, we show that Phi commutes with the virtualization map and that B-1,B-s is naturally a virtual crystal in type D-4((1)), thus defining an affine crystal structure on rigged configurations corresponding to B-1,B-s. (C) 2015 Elsevier Inc. All rights reserved
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Embeddings of right-angled Artin groups
No full textWe explicitly construct an embedding of a right-angled Artin group into a classical
pure braid group. Using this we obtain a number of corollaries describing embeddings of
arbitrary Artin groups into right-angled Artin groups and linearly independent subgroups of
a right-angled Artin group
Free fermionic probability theory and K-theoretic Schubert calculus
No full textFor each of the four particle processes given by Dieker and Warren
[arXiv:0707.1843], we show the -step transition kernels are given by the
(dual) (weak) refined symmetric Grothendieck functions up to a simple overall
factor. We do so by encoding the particle dynamics as the basis of free
fermions first introduced by the first author, which we translate into deformed
Schur operators acting on partitions. We provide a direct combinatorial proof
of this relationship in each case, where the defining tableaux naturally
describe the particle motions.Comment: 52 pages, 5 figures, 2 table
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