34,945 research outputs found
Non-commutative Pieri operators on posets
We consider graded representations of the algebra NC of noncommutative
symmetric functions on the Z-linear span of a graded poset P. The matrix
coefficients of such a representation give a Hopf morphism from a Hopf algebra
HP generated by the intervals of P to the Hopf algebra of quasi-symmetric
functions. This provides a unified construction of quasi-symmetric generating
functions from different branches of algebraic combinatorics, and this
construction is useful for transferring techniques and ideas between these
branches. In particular we show that the (Hopf) algebra of Billera and Liu
related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge
related to enriched P-partitions, and connect this to the combinatorics of the
Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete
and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to
G.-C. Rot
Invariant tensors and the cyclic sieving phenomenon
We construct a large class of examples of the cyclic sieving phenomenon by
expoiting the representation theory of semi-simple Lie algebras. Let be a
finite dimensional representation of a semi-simple Lie algebra and let be
the associated Kashiwara crystal. For , the triple which
exhibits the cyclic sieving phenomenon is constructed as follows: the set
is the set of isolated vertices in the crystal ; the map is a generalisation of promotion acting on standard tableaux of
rectangular shape and the polynomial is the fake degree of the Frobenius
character of a representation of related to the natural action
of on the subspace of invariant tensors in .
Taking to be the defining representation of gives the
cyclic sieving phenomenon for rectangular tableaux
The biHecke monoid of a finite Coxeter group
The usual combinatorial model for the 0-Hecke algebra of the symmetric group
is to consider the algebra (or monoid) generated by the bubble sort operators.
This construction generalizes to any finite Coxeter group W. The authors
previously introduced the Hecke group algebra, constructed as the algebra
generated simultaneously by the bubble sort and antisort operators, and
described its representation theory.
In this paper, we consider instead the monoid generated by these operators.
We prove that it has |W| simple and projective modules. In order to construct a
combinatorial model for the simple modules, we introduce for each w in W a
combinatorial module whose support is the interval [1,w] in right weak order.
This module yields an algebra, whose representation theory generalizes that of
the Hecke group algebra. This involves the introduction of a w-analogue of the
combinatorics of descents of W and a generalization to finite Coxeter groups of
blocks of permutation matrices.Comment: 12 pages, 1 figure, submitted to FPSAC'1
Overview of the Heisenberg--Weyl Algebra and Subsets of Riordan Subgroups
In a first part, we are concerned with the relationships between polynomials
in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock
representation with differential operators and the associated one-parameter
group.Upon this basis, the paper is then devoted to the groups of Riordan
matrices associated to the related transformations of matrices (i.e.
substitutions with prefunctions). Thereby, various properties are studied
arising in Riordan arrays, in the Riordan group and, more specifically, in the
`striped' Riordan subgroups; further, a striped quasigroup and a semigroup are
also examined. A few applications to combinatorial structures are also briefly
addressed in the Appendix.Comment: Version 3 of the paper entitled `On subsets of Riordan subgroups and
Heisenberg--Weyl algebra' in [hal-00974929v2]The present article is published
in The Electronic Journal of Combinatorics, Volume 22, Issue 4, 40 pages
(Oct. 2015), pp.Id: 1
The geometry and combinatorics of Springer fibers
This survey paper describes Springer fibers, which are used in one of the
earliest examples of a geometric representation. We will compare and contrast
them with Schubert varieties, another family of subvarieties of the flag
variety that play an important role in representation theory and combinatorics,
but whose geometry is in many respects simpler. The end of the paper describes
a way that Springer fibers and Schubert varieties are related, as well as open
questions.Comment: 18 page
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