134 research outputs found
Nilpotency in type A cyclotomic quotients
We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree
of cyclotomic quotients of rings that categorify one-half of quantum sl(k).Comment: 19 pages, 39 eps files. v3 simplifies antigravity moves and corrects
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Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule
The basic representation of \A is studied. The weight vectors are
represented in terms of Schur functions. A suitable base of any weight space is
given. Littlewood-Richardson rule appears in the linear relations among weight
vectors.Comment: February 1995, 7pages, Using AMS-Te
Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
We construct an explicit isomorphism between blocks of cyclotomic Hecke
algebras and (sign-modified) Khovanov-Lauda algebras in type A. These
isomorphisms connect the categorification conjecture of Khovanov and Lauda to
Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally
graded, which allows us to exhibit a non-trivial Z-grading on blocks of
cyclotomic Hecke algebras, including symmetric groups in positive
characteristic.Comment: 32 pages; minor changes to section
Compound basis arising from the basic -module
A new basis for the polynomial ring of infinitely many variables is
constructed which consists of products of Schur functions and Q-functions. The
transition matrix from the natural Schur function basis is investigated.Comment: 12 page
On the Representation Theory of an Algebra of Braids and Ties
We consider the algebra introduced by F. Aicardi and J.
Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor
space representation for and show that this is faithful. We use
it to give a basis for and to classify its irreducible
representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic
Combinatorics
The degenerate analogue of Ariki's categorification theorem
We explain how to deduce the degenerate analogue of Ariki's categorification
theorem over the ground field C as an application of Schur-Weyl duality for
higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also
discuss some supplementary topics, including Young modules, tensoring with
sign, tilting modules and Ringel duality.Comment: 44 page
Combinatorial Hopf algebras and Towers of Algebras
Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
gives rise to graded dual Hopf algebras then we must
have where .Comment: 7 page
The Symbolic and cancellation-free formulae for Schur elements
In this paper we give a symbolical formula and a cancellation-free formula
for the Schur elements associated to the simple modules of the degenerate
cyclotomic Hecke algebras. As some direct applications, we show that the Schur
elements are symmetric with respect to the natural symmetric group action and
are integral coefficients polynomials and we give a different proof of
Ariki-Mathas-Rui's criterion on the semi-simplicity of degenerate cyclotomic
Hecke algebras.Comment: To appear in Monatshefte fur Mathemati
New reflection matrices for the U_q(gl(m|n)) case
We examine super symmetric representations of the B-type Hecke algebra. We
exploit such representations to obtain new non-diagonal solutions of the
reflection equation associated to the super algebra U_q(gl(m|n)). The boundary
super algebra is briefly discussed and it is shown to be central to the super
symmetric realization of the B-type Hecke algebraComment: 13 pages, Latex. A few alterations regarding the representations. A
reference adde
Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
The box-ball system is an integrable cellular automaton on one dimensional
lattice. It arises from either quantum or classical integrable systems by the
procedures called crystallization and ultradiscretization, respectively. The
double origin of the integrability has endowed the box-ball system with a
variety of aspects related to Yang-Baxter integrable models in statistical
mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz,
geometric crystals, classical theory of solitons, tau functions, inverse
scattering method, action-angle variables and invariant tori in completely
integrable systems, spectral curves, tropical geometry and so forth. In this
review article, we demonstrate these integrable structures of the box-ball
system and its generalizations based on the developments in the last two
decades.Comment: 73 page
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