Results 1  10
of
16
Combinatorial Hopf algebras
"... Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebr ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We give a precise definition of “combinatorial Hopf algebras”, and we classify them in the four cases: associative or commutative, general or rightsided. For instance a cofreecocommutative combinatorial Hopf algebra is completely determined by its primitive part which is a preLie algebra. The classification gives rise to several good triples of operads. It involves the operads: dendriform, preLie, brace, GerstenhaberVoronov, and variations of them.
A jeu de taquin theory for increasing tableau, with applications to Ktheoretic Schubert calculus
, 2007
"... We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Schützenberger ’77] for standard Young tableaux. We apply this to give a new combinatorial rule for the Ktheory Schubert calculus of Grassmannians via Ktheoretic jeu de taquin, providing an alternative ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Schützenberger ’77] for standard Young tableaux. We apply this to give a new combinatorial rule for the Ktheory Schubert calculus of Grassmannians via Ktheoretic jeu de taquin, providing an alternative to the rules of [Buch ’02] and others. This rule naturally generalizes to give a conjectural rootsystem uniform rule for any minuscule flag variety G/P, extending [ThomasYong ’06]. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.
Ktheory Schubert calculus of the affine Grassmannian
, 2009
"... We construct the Schubert basis of the torusequivariant Khomology of the affine Grassmannian of a simple algebraic group G, using the Ktheoretic NilHecke ring of Kostant and Kumar. This is the Ktheoretic analogue of a construction of Peterson in equivariant homology. For the case G = SLn, the K ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
We construct the Schubert basis of the torusequivariant Khomology of the affine Grassmannian of a simple algebraic group G, using the Ktheoretic NilHecke ring of Kostant and Kumar. This is the Ktheoretic analogue of a construction of Peterson in equivariant homology. For the case G = SLn, the Khomology of the affine Grassmannian is identified with a subHopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called KkSchur functions, whose highest degree term is a kSchur function. The dual basis in Kcohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in Khomology. Many of our constructions have geometric interpretations using Kashiwara’s thick affine flag manifold.
Tridendriform structure on combinatorial Hopf algebras
 J. Algebra
, 2010
"... Abstract. We extend the definition of tridendriform bialgebra by introducing a parameter q. The subspace of primitive elements of a qtridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q Gerst ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We extend the definition of tridendriform bialgebra by introducing a parameter q. The subspace of primitive elements of a qtridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q GerstenhaberVoronov algebras. We prove the equivalence between the categories of conilpotent qtridendriform bialgebras and of qGerstenhaberVoronov algebras. The space spanned by surjective maps between finite sets, as well as the space spanned by parking functions, have a natural structure of qtridendriform bialgebra, denoted ST(q) and PQSym(q)∗, in such a way that ST(q) is a subtridendriform bialgebra of PQSym(q)∗. Finally we show that the bialgebra of Mpermutations defined by T. Lam and P. Pylyavskyy comes from a qtridendriform algebra which is a quotient of ST(q).
The direct sum map on Grassmannians and jeu de taquin for increasing tableaux
 Intern. Math. Res. Notices
"... ar ..."
(Show Context)
Combinatorics of the Ktheory of affine Grassmannians, preprint arXiv:0907.0044
 THOMAS LAM, ANNE SCHILLING, AND MARK SHIMOZONO
"... Abstract. We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and kSchur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and kKSchur functions – Schubert representat ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and kSchur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and kKSchur functions – Schubert representatives for the Ktheory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules. 1. Introduction and
Hopf algebras in combinatorics
, 2013
"... Certain Hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects (partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc). The rigidity in the structure of a Hopf algebra can lead to enlightening proofs, and many i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Certain Hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects (partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc). The rigidity in the structure of a Hopf algebra can lead to enlightening proofs, and many interesting invariants of combinatorial objects turn out to be evaluations of Hopf morphisms. These are lecture notes for Fall 2012 Math 8680 Topics in Combinatorics at the University of Minnesota. The course is an attempt to focus on examples that I find interesting, but which are hard to find fully explained currently in books or in one paper. Be warned that these notes are highly idiosyncratic in choice
1 On the JacobiTrudi formula for dual stable Grothendieck
"... We will first begin with a review of the facts that we already now about this problem. Firstly, a semistandard Young tableau T is a Young tableau λ = (λ1,..., λm) with positive integer entries which strictly increase in columns and weakly increase in rows. Secondly, we will define a Schur function: ..."
Abstract
 Add to MetaCart
(Show Context)
We will first begin with a review of the facts that we already now about this problem. Firstly, a semistandard Young tableau T is a Young tableau λ = (λ1,..., λm) with positive integer entries which strictly increase in columns and weakly increase in rows. Secondly, we will define a Schur function: a Schur function is a polynomial sλ is defined as sλ = ∑ x T = ∑ x t1 1 xt2 2 · · · xtn n, (1) T T where the summation is over all semistandard Young tableau T of shape λ; the exponents t1,..., tn represent the weight of the tableau, in other words the ti counts the number of occurences of i in T. Thirdly, a reverse plane partition is Young tableau with positive integer entries which increase weakly both in rows and columns. Forthly,we introduce the dualstable Grothendieck polynomials, defined as gλ = ∑ xT = ∑ x t1
HOOK FORMULAS FOR SKEW SHAPES
"... Abstract. The celebrated hooklength formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hookle ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The celebrated hooklength formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hooklengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functions and a generalization of the HillmanGrassl correspondence, respectively. Our main results are two qanalogues of Naruse’s formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations. In particular, we obtain new summation formulas for the number of alternating permutations in terms of certain Dyck paths, and their qanalogues. 1.