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Mike Zabrocki

Publications and source records attributed to Mike Zabrocki.

52 records · Page 3Linked to original sources
arXiv2008

Analytic aspects of the shuffle product

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models D-finite generating functions.

cs.IT↗
arXiv2007

On the S_n-module structure of the noncommutative harmonics

Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.

math.CO↗
arXiv2006

Posets related to the connectivity set of Coxeter groups

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies that any interval is Cohen-Macauley. We also give a Galois connection between intervals in this poset and a boolean poset. This allows us to compute the Mobius function for any interval.

math.CO↗
arXiv2005

The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

math.CO↗
arXiv2005

Grothendieck bialgebras, Partition lattices and symmetric functions in noncommutative variables

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

math.CO↗
arXiv2005

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions indexed by compositions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials.

math.CO↗
arXiv2004

Deformed universal characters for classical and affine algebras

Creation operators are given for the three distinguished bases of the type BCD universal character ring of Koike and Terada yielding an elegant way of treating computations for all three types in a unified manner. Deformed versions of these operators create symmetric function bases whose expansion in the universal character basis, has polynomial coefficients in $q$ with non-negative integer coefficients. We conjecture that these polynomials are one-dimensional sums associated with crystal bases of finite-dimensional modules over quantized affine algebras for all nonexceptional affine types. These polynomials satisfy a Macdonald-type duality.

math.CO↗
arXiv2003

A bijective proof of an unusual symmetric group generating function

For $σ\in S_n$, let $D(σ) = \{i : σ_{i} > σ_{i+1}\}$ denote the descent set of $σ$. The length of the permutation is the number of inversions, denoted by $inv(σ) = \big | \{(i,j) : i σ_j\} \big |$. Define an unusual quadratic statisitic by $baj(σ) = \sum_{i \in D(σ)} i (n-i)$. We present here a bijective proof of the identity $\sum_{{σ\in S_n} \atop {σ(n) = k}} q^{baj(σ) - inv(σ)} = \prod_{i=1}^{n-1} {1-q^{i (n-i)} \over {1-q^i}}$ where $k$ is a fixed integer.

math.CO↗
arXiv2002

q-Analogs of symmetric function operators

For any homomorphism V on the space of symmetric functions, we introduce an operation which creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall-Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q,t-Kostka coefficients.

math.QA↗
arXiv2000

Polynomiality of the q,t-Kostka Revisited

Let $K(q,t)= \|K_{\laμ}(q,t)\|_{\la,μ}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\laμ}(t). In this paper we present a new proof of the polynomiality of the q,t-Kostka coefficients that is both short and elementary. More precisely, we derive that $K(q,t)$ has entries in \ZZ[q,t] directly from the fact that the matrix $K(t)^{-1}$ has entries in \ZZ[t]. The proof uses only identities that can be found in the original paper [7] of Macdonald.

math.QA↗
arXiv2000

Ribbon Operators and Hall-Littlewood Symmetric Functions

Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities.

math.QA↗
arXiv2000

Hall-Littlewood vertex operators and generalized Kostka polynomials

A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.

math.QA↗
arXiv1999

Positivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics

We present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{3}^{qt} H_{(2^a1^b)}[X;q,t] = H_{(32^a1^b)}[X;q,t]$ and $H_4^{qt} H_{(2^a1^b)}[X;q,t] = H_{(42^a1^b)}[X;q,t]$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_μ(T)$ and $b_μ(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\laμ}(q,t) = \sum_T t^{a_μ(T)} q^{b_μ(T)}$ for the case when when $μ$ is two columns or of the form $(32^a1^b)$ or $(42^a1^b)$. This provides proof of the positivity of the $(q,t)$-Kostka coefficients in the previously unknown cases of $K_{\la (32^a1^b)}(q,t)$ and $K_{\la (42^a1^b)}(q,t)$. The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when $μ$ is two columns.

math.CO↗
arXiv1999

Vertex operators for standard bases of the symmetric functions

We present a formulas to add a row or a column to the power, monomial, forgotten, Schur, homogeneous and elementary symmetric functions. As an application of these operators we show that the operator that adds a column to the Schur functions can be used to calculate a formula for the number of pairs of standard tableaux the same shape and height less than or equal to a fixed $k$.

math.CO↗