Searcharxiv⌕ Search

arXiv subjects

Mike Zabrocki

Publications and source records attributed to Mike Zabrocki.

At least 37 records · Page 2Linked to original sources
arXiv2017

Products of characters of the symmetric group

In a recent paper, the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products that have as consequences several versions of the Pieri rule for this new basis of symmetric functions. In addition, we give several applications of the products studied in this paper.

math.CO↗
arXiv2016

Symmetric group characters as symmetric functions (extended abstract)

The irreducible characters of the symmetric group are a symmetric polynomial in the eigenvalues of a permutation matrix. They can therefore be realized as a symmetric function that can be evaluated at a set of variables and form a basis of the symmetric functions. This basis of the symmetric functions is of non-homogeneous degree and the (outer) product structure coefficients are the stable Kronecker coefficients. We introduce the irreducible character basis by defining it in terms of the induced trivial characters of the symmetric group which also form a basis of the symmetric functions. The irreducible character basis is closely related to character polynomials and we obtain some of the change of basis coefficients by making this connection explicit. Other change of basis coefficients come from a representation theoretic connection with the partition algebra, and still others are derived by developing combinatorial expressions. This document is an extended abstract which can be used as a review reference so that this basis can be implemented in Sage. A more complete version of the results in this abstract can be found in {\tt arXiv:1605.06672}.

math.CO↗
arXiv2016

Multiplicative structures of the immaculate basis of non-commutative symmetric functions

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.

math.CO↗
arXiv2014

Symmetries on the lattice of $k$-bounded partitions

In 2002 R. Suter identified a dihedral symmetry on certain order ideals in Young's lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be seen to be in bijection with the alcoves in a 2-fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by L. Lapointe, A. Lascoux, and J. Morse in the study of k-Schur functions. We identify the order ideal and the cyclic action on it explicitly in a geometric and combinatorial form.

math.CO↗
arXiv2014

The Pieri rule for dual immaculate quasi-symmetric functions

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third author to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.

math.CO↗
arXiv2013

$k$-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

math.CO↗
arXiv2013

A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

math.CO↗
arXiv2013

A three shuffle case of the compositional parking function conjecture

We prove here that the polynomial q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and have a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp(1) is a rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator introduced in [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of [14]. It includes as special cases all previous results connected with this conjecture such as the q, t-Catalan [3] and the Schroder and h, h results of Haglund in [12] as well as their compositional refinements recently obtained in [9] and [10]. It also confirms the possibility that the approach adopted in [9] and [10] has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki in [15].

math.CO↗
arXiv2013

The immaculate basis of the non-commutative symmetric functions

FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.

math.CO↗
arXiv2012

A new `dinv' arising from the two part case of the Shuffle Conjecture

In a recent paper J. Haglund showed that a certain symmetric function expresion enumerates by t^{area} q^{dinv} of the parking functions whose diagonal word is in the shuffle of 12...j and j+1...j+n with k of the cars j+1,...,j+n in the main diagonal including car j+n in the cell (1,1). In view of some recent conjectures of Haglund-Morse-Zabrocki it is natural to conjecture that replacing E_{n,k} by the modified Hall-Littlewood functions would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting cars j+1,...,j+n hits the diagonal according to the composition p=(p_1,p_2,...,p_k). We prove here this conjecture by deriving a recursion for the symmetric function expression then using this recursion to construct a new dinv statistic we will denote ndinv and show that this polynomial enumerates the latter parking functions by t^{area} q^{ndinv}.

math.CO↗
arXiv2011

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

math.CO↗
arXiv2011

A compositional shuffle conjecture specifying touch points of the Dyck path

We introduce a $q,t$-enumeration of Dyck paths which are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall-Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. $\mathbf {126}$ (2005), pp. 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences and we prove a number of identities involving these functions.

math.CO↗
arXiv2011

The Murnaghan-Nakayama rule for k-Schur functions

We prove the Murgnaghan--Nakayama rule for $k$-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a $k$-Schur function in terms of $k$-Schur functions. This is proved using the noncommutative $k$-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.

math.CO↗
arXiv2010

Words and polynomial invariants of finite groups in non-commutative variables

Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x_1, x_2,..., x_n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words.

math.CO↗
arXiv2008

Expressions for Catalan Kronecker Products

We give some elementary manifestly positive formulae for the Kronecker products s_(d,d) * s_(d+k,d-k) and s_(d,d) * s_(2d-k,1^k). These formulae demonstrate some fundamental properties of the Kronecker coefficients, and we use them to deduce a number of enumerative and combinatorial results.

math.CO↗