260 research outputs found
Free shuffle algebras in language varieties
AbstractWe give simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” LΣ := (P(Σ∗),+,., ⊗, 0,1), or the semirings RΣ := (R(Σ∗),+,., ⊗,∗,0,1), where P(Σ∗) is the collection of all subsets of the free monoid Σ∗, and R(Σ∗) is the collection of all regular subsets. The operation x ⊗ y is the shuffle product
The Feigin Tetrahedron
The first goal of this note is to extend the well-known Feigin homomorphisms
taking quantum groups to quantum polynomial algebras. More precisely, we define
generalized Feigin homomorphisms from a quantum shuffle algebra to quantum
polynomial algebras which extend the classical Feigin homomorphisms along the
embedding of the quantum group into said quantum shuffle algebra. In a recent
work of Berenstein and the author, analogous extensions of Feigin homomorphisms
from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial
algebras were defined. To relate these constructions, we establish a
homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel
algebra to the quantum shuffle algebra which relates the generalized Feigin
homomorphisms. These constructions can be compactly described by a commuting
tetrahedron of maps beginning with the quantum group and terminating in a
quantum polynomial algebra. The second goal in this project is to better
understand the dual canonical basis conjecture for skew-symmetrizable quantum
cluster algebras. In the symmetrizable types it is known that dual canonical
basis elements need not have positive multiplicative structure constants, while
this is still suspected to hold for skew-symmetrizable quantum cluster
algebras. We propose an alternate conjecture for the symmetrizable types: the
cluster monomials should correspond to irreducible characters of a KLR algebra.
Indeed, the main conjecture of this note would establish this "KLR conjecture"
for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture
that the images of rigid representations under the quantum shuffle character
give irreducible characters for KLR algebras
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
Pattern avoidance in labelled trees
We discuss a new notion of pattern avoidance motivated by the operad theory:
pattern avoidance in planar labelled trees. It is a generalisation of various
types of consecutive pattern avoidance studied before: consecutive patterns in
words, permutations, coloured permutations etc. The notion of Wilf equivalence
for patterns in permutations admits a straightforward generalisation for (sets
of) tree patterns; we describe classes for trees with small numbers of leaves,
and give several bijections between trees avoiding pattern sets from the same
class. We also explain a few general results for tree pattern avoidance, both
for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining
the operadic contex
Asymmetric function theory
The classical theory of symmetric functions has a central position in
algebraic combinatorics, bridging aspects of representation theory,
combinatorics, and enumerative geometry. More recently, this theory has been
fruitfully extended to the larger ring of quasisymmetric functions, with
corresponding applications. Here, we survey recent work extending this theory
further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the
Schubert calculus conference in Guangzhou, Nov. 201
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