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We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative sub-algebra of $R_n$. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,$$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition, Comment: 17 pages, minor corrections to paper, including remarks from Darij Grinberg

We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset tableau of content $[k]$. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson., Comment: 30 pages

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that has bases elements indexed by multiset partitions, where $x$ is an indeterminate and $r$ and $k$ are non-negative integers. This algebra can be realized as a diagram algebra that generalizes the partition algebra. When $x$ is an integer greater or equal to $2r$, we show that ${\rm M\!P}_{r,k}(x)$ is isomorphic to a centralizer algebra of the symmetric group, $S_n$, acting on the polynomial ring on the variables $x_{ij}$, $1\leq i \leq n$ and $1\leq j\leq k$. We describe the representations of ${\rm M\!P}_{r,k}(x)$, branching rule and restriction of its representations in the case that $x$ is an integer greater or equal to $2r$., 24 pages

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis., arXiv admin note: text overlap with arXiv:1605.06672; remark 1: irreducible character basis of the rook algebra

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., Comment: new version includes edits, references to forthcoming research, and a 'hook-length' formula for the number of standard immaculate tableaux

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]., Comment: 30 pages

In The Delta Conjecture (arXiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $\Delta_{h_\ell} \nabla C_\alpha$ is equal to a weighted sum over decorated Dyck paths.

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis., 35 pages; this is the more complete version of arXiv:1510.00438; v5 differs from previous versions with minor edits and the addition of an appendix about using Sagemath to do computations with these bases (this appendix does not appear in the published journal version)

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra., Comment: to appear in Journal of Combinatorics, 28 pages

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., Comment: 13 pages. Expanded version of a paper to appear in "Journal of Combinatorial Theory, series A" (http://www.elsevier.com/locate/jcta)

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., 29 pages

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., Comment: 17 pages - minor revisions to appear in Discrete Mathematics issue for LaCIM'2000

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., 14 pages, corrections from v1 and v2

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., 30 pages: we cleaned and fixed many details in the proofs. The interested reader may toggle \specialcomments in the TeX file to reveal 6 added pages of details and ideas (in red)

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}., 32 pages, 28 figures

We prove a Murnaghan–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., Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

We give a type free formula for the expansion of $k$-Schur functions indexed by fundamental coweights within the affine nilCoxeter algebra. Explicit combinatorics are developed in affine type $C$.

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., 20 pages

Let $V$ be a complex vector space with basis $\{x_1,x_2,\ldots,x_n\}$ 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 $x_1, x_2, \ldots, x_n$ 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 $G$-module in terms of words in a particular Cayley graph of $G$. To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of 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., Soit V un espace vectoriel complexe de base $\{x_1,x_2,\ldots,x_n\}$ et $G$ un sous-groupe fini de $GL(V)$. L'algèbre $T(V)$ des tenseurs de $V$ sur les complexes est isomorphe aux polynômes à coefficients complexes en variables non-commutatives $x_1, x_2, \ldots, x_n$. Nous voulons donner une décomposition de $T(V)$ en $G$-modules simples de manière combinatoire. Plus particulièrement, nous étudions l'espace gradué des invariants de $T(V)$ sous l'action de $G$. Nous présentons une méthode générale donnant la décomposition de $T(V)$ en modules simples via certains mots dans un graphe de Cayley donné. Pour appliquer la méthode à un groupe particulier, nous avons besoin d'un homomorphisme surjectif entre une sous-algèbre de l'algèbre de groupe et l'algèbre des caractères. Pour le cas du groupe symétrique, nous donnons un exemple de cet homomorphisme qui provient de la théorie de l'algèbre des descentes. Pour le groupe diédral, nous avons une réalisation de l'algèbre des caractères comme une sous-algèbre de l'algèbre de groupe. Dans ces deux cas, nous avons une interprétation des dimensions graduées de l'espace des invariants en terme de ces mots.

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., Comment: 19 pages

We introduce non-commutative analogs of k-Schur functions of Lapointe–Lascoux and Morse. We give explicit formulas for the expansions of non-commutative functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case.

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., 17 pages

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., Comment: 22 pages; 9 figures in eps form

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., Comment: 30 pages

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, $a_{\mu}(T)$ and $b_{\mu}(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\la\mu}(q,t) = \sum_T t^{a_{\mu}(T)} q^{b_{\mu}(T)}$ for the case when when $\mu$ 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 $\mu$ is two columns., Comment: LaTEX, 37 pages, Replacement of submission with vertex operators only

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., Comment: To Appear in Advances in Mathematics (2012), 23 pages

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., Comment: 13 pages; 4 figures; final version; accepted in Journal of Algebra

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.

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.