2,206 research outputs found
Combinatorial Hopf algebra of superclass functions of type
We provide a Hopf algebra structure on the space of superclass functions on
the unipotent upper triangular group of type D over a finite field based on a
supercharacter theory constructed by Andr\'e and Neto. Also, we make further
comments with respect to types B and C. Type A was explores by M. Aguiar et. al
(2010), thus this paper is a contribution to understand combinatorially the
supercharacter theory of the other classical Lie types.Comment: Last section modified. Recent development added and correction with
respect to previous version state
Combinatorial Representation Theory
We attempt to survey the field of combinatorial representation theory,
describe the main results and main questions and give an update of its current
status. We give a personal viewpoint on the field, while remaining aware that
there is much important and beautiful work that we have not been able to
mention
Vertex operators arising from Jacobi-Trudi identities
We give an interpretation of the boson-fermion correspondence as a direct
consequence of Jacobi-Trudi identity. This viewpoint enables us to construct
from a generalized version of the Jacobi-Trudi identity the action of Clifford
algebra on polynomial algebras that arrives as analogues of the algebra of
symmetric functions. A generalized Giambelli identity is also proved to follow
from that identity. As applications, we obtain explicit formulas for vertex
operators corresponding to characters of the classical Lie algebras, shifted
Schur functions, and generalized Schur symmetric functions associated to linear
recurrence relations.Comment: 23 page
Hopf monoids from class functions on unitriangular matrices
We build, from the collection of all groups of unitriangular matrices, Hopf
monoids in Joyal's category of species. Such structure is carried by the
collection of class function spaces on those groups, and also by the collection
of superclass function spaces, in the sense of Diaconis and Isaacs.
Superclasses of unitriangular matrices admit a simple description from which we
deduce a combinatorial model for the Hopf monoid of superclass functions, in
terms of the Hadamard product of the Hopf monoids of linear orders and of set
partitions. This implies a recent result relating the Hopf algebra of
superclass functions on unitriangular matrices to symmetric functions in
noncommuting variables. We determine the algebraic structure of the Hopf
monoid: it is a free monoid in species, with the canonical Hopf structure. As
an application, we derive certain estimates on the number of conjugacy classes
of unitriangular matrices.Comment: Final Version, 32 pages, accepted in "Algebra and Number Theory
Integrable Combinatorics
We review various combinatorial problems with underlying classical or quantum
integrable structures. (Plenary talk given at the International Congress of
Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1
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