164 research outputs found
Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued
Richard Stanley played a crucial role, through his work and his students, in
the development of the relatively new area known as combinatorial
representation theory. In the early stages, he has the merit to have pointed
out to combinatorialists the potential that representation theory has for
applications of combinatorial methods. Throughout his distinguished career, he
wrote significant articles which touch upon various combinatorial aspects
related to representation theory (of Lie algebras, the symmetric group, etc.).
I describe some of Richard's contributions involving Lie algebras, as well as
recent developments inspired by them (including some open problems), which
attest the lasting impact of his work.Comment: 11 page
Super quasi-symmetric functions via Young diagrams
We consider the multivariate generating series of -partitions in
infinitely many variables . For some family of ranked posets
, it is natural to consider an analog with two infinite alphabets.
When we collapse these two alphabets, we trivially recover . Our main
result is the converse, that is, the explicit construction of a map sending
back onto . We also give a noncommutative analog of the latter. An
application is the construction of a basis of WQSym with a non-negative
multiplication table, which lifts a basis of QSym introduced by K. Luoto.Comment: 12 pages, extended abstract of arXiv:1312.2727, presented at FPSAC
conference. The presentation of the results is quite different from the long
versio
Flag-symmetry of the poset of shuffles and a local action of the symmetric group
We show that the poset of shuffles introduced by Greene in 1988 is
flag-symmetric, and we describe a "local" permutation action of the symmetric
group on the maximal chains which is closely related to the flag symmetric
function of the poset. A key tool is provided by a new labeling of the maximal
chains of a poset of shuffles, which is also used to give bijective proofs of
enumerative properties originally obtained by Greene. In addition we define a
monoid of multiplicative functions on all posets of shuffles and describe this
monoid in terms of a new operation on power series in two variables.Comment: 34 pages, 6 figure
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