Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is the n-th elementary symmetric function, and \nabla is an operator from the theory of Macdonald polynomials.
We conjecture a combinatorial formula for \nabla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on \nabla e_n are special cases of our conjecture.
Finally, we extend our conjectures on \nabla e_n and several of the results supporting them to higher powers \nabla^m e_n.
We describe a general conjecture on how one may derive from the generic bosonic case all structural properties of multivariate diagonal coinvariant modules in $k$ sets of $n$ commuting variables (bosons), and $j$ sets of $n$ anticommuting variables (fermions).
We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category $\cO$ for the rational Cherednik algebra of $W$.
We explain how to use representation theory to give a lower bound on the dimension of the quotient ring by type $B_n$ diagonal invariants that improves upon the current known lower bound $(2n+1)^n$ by a quadratic polynomial in $n$.
The purpose of this paper is to give an explicit description of the irreducible decomposition of the multigraded S_n-module of coinvariants of S_n x S_n. Many of the results presented can be extended to analogous questions for other finite reflection group.
For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of $r$.
We calculate the Borel-Moore homology of affine Springer fibers of type $A$ associated to some regular semisimple nil elliptic elements. As a result, we obtain bigraded $\mf{S}_{n}$-modules whose bigraded Frobenius series are generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.
Let $Θ_n = (θ_1, \dots, θ_n)$ and $Ξ_n = (ξ_1, \dots, ξ_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ Θ_n, Ξ_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ Θ_n, Ξ_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via n…
Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of…
Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. We give an explicit description of these actions, and deduce a combinatorial formula for the resulting graded characters on the coinvariant algebra.
Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$ of the polynomial ring $\mathbb{C}[x_1, \dots, x_n]$ in $n$ variables. When $k = n$, both of these quotients coincide with the classical coinvariant algebra attached to $G_n$. The algebraic properties of our quotients are governed by the combinatorial properties of $k$-dimensional faces in the Coxeter complex attached to $G_n$ (in the case of $R_{n,k}$) and…
Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\em tail positive words}. We calcul…
We introduce the family of Theta operators $Θ_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson for $Δ_{e_{n-k-1}}'e_n$. We show that the $4$-variable Catalan theorem of Zabrocki is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit, extending one of their results. Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for $Δ_{h_m}Δ_{…
In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of $\operatorname{DR}_n$ which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the first author, Remmel and Wilson (2018)…
The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant polynomials with vanishing constant term. The quotient $R_n = \frac{\mathbb{Q}[\mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} \subseteq \mathbb{Q}[\mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $…
In the context of the ring Q[x,y], of polynomials in 2n variables x=x1,...,x_n and y=y1,...,yn, we introduce the notion of diagonally quasi-symmetric polynomials. These, also called "diagonal Temperley-Lieb invariants", make possible the further introduction of the space of "diagonal Temperley-Lieb harmonics" and "diagonal Temperley-Lieb coinvariant space". We present new results and conjectures concerning these spaces, as well as the space obtained as the quotient of the ring of diagonal Temperley-Lieb invariants by the ideal generated by constant term free diagonally symmetric invariants. We…
This paper's aim is to present recent combinatorial considerations on r-Dyck paths, r-Parking functions, and the r-Tamari lattices. Giving a better understanding of the combinatorics of these objects has become important in view of their (conjectural) role in the description of the graded character of the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the symmetric group.
In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\sf area'} and {\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that…
We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative results follow and allow to look at the sub-class of the so- called bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type tableaux and coloring of hypergraphs having a square number of vertices.
We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type.
Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.
We give an explicit combinatorial formula for some irreducible components of $GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To this end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give the formula directly in terms of Schur functions. This paper also contains formulas written in terms of Schur functions in the $q$ and $t$ variables for special cases of $\nabla(e_n)$, $\nabla^r(e_n)$ and $Δ'_{e_k}(e_n)$. We also give an interpretation in term of path to the adjoint dual Pieri rule applied on these $GL_k\times \mathbb{S}_n$-characters.
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues of cluster mutations for polytopes, of matching field polytopes and show that the property of giving rise to a toric degeneration of the Grassmannians, is preserved by mutation. Moreover the polyto…
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.
We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tildeÅ$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conju…
We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.
A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is further investigated in this paper and extended to the hyperoctahedral group $B_n$. Key ingredients are a new formula for the irreducible characters of $B_n$, the signed quasisymmetric functions introduced by Poirier, and a new family of matrices of Walsh--Hadamard type. Applications include formulas for natural $B_n$-actions on coinvariant and exterior algebras…
We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.
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 a…
We consider the graded $§_n$-modules of higher diagonally harmonic polynomials in three sets of variables (the trivariate case), and show that they have interesting ties with generalizations of the Tamari poset and parking functions. In particular we get several nice formulas for the associated Hilbert series and graded Frobenius characteristic. This also leads to entirely new combinatorial formulas.
In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=α$ over the finite field $\mathbb{F}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs.
Experimental mathematics is an experimental approach to mathematics in which programming and symbolic computation are used to investigate mathematical objects, identify properties and patterns, discover facts and formulas and even automatically prove theorems. With an experimental mathematics approach, this dissertation deals with several combinatorial problems and demonstrates the methodology of experimental mathematics. We start with parking functions and their moments of certain statistics. Then we discuss about spanning trees and "almost diagonal" matrices to illustrate the methodology of …
To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called "diagonal basis". We prove that this definition extends naturally to a large class of algebras, the so called chi-algebras. Our definitions make also use of an "iterative residue formula" based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal…
We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the $\mathbb{Z}$-linear relations between small $m$-diagonals $\text{pt}^{\{1,\ldots,n\}\setminus A}\timesΔ_A$ in the rational Chow ring of $X^n$ for $A$ ranging over $m$-element subsets of $\{1,\ldots,n\}$. Our results generalize to arbitrary symmetric classes in place of the diagonal in $X^m$, and with different types of inclusions $A^\bullet(X^m)_{\mathbb{Q}}^{S_m} \hookrightarrow A^\bullet(X^n)_{\mathbb{Q}}$. The combinatorial heart of this paper, which may be of independent interest, is showing the $\m…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.
In this paper, we generalize Haagerup's inequality (on convolution norm in the free group) to a very general context of R-diagonal elements in a tracial von Neumann algebra; moreover, we show that in this "holomorphic" setting, the inequality is greatly improved from its originial form. We give an elementary combinatorial proof of a very special case of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for R-diagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultra…
Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, the distri- bution of parameters on the first diagonal was proven to be asymptotically normal. In that same paper, a conjecture was made that the other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds up…
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic which is composed of three parts and which is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a geometric description of the statistic in the style of the classical case. We go on to show that in the n x (n+1) lattice, parki…
Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize diagonal solutions and use this to prove that a special family of formal power series, called content-type series, are solutions to the 2-Toda hierarchy. As examples, we prove that various generating series for permutation factorization problems (including the double Hurwitz problem), correlators for the Schur measure on partitions and the formal character ex…
The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters.
Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of vertices in any minimal vertex cover of C. Under certain conditions we prove that C is vertex critical. If C satisfies the max-flow min-cut property, we prove that A diagonalizes over the integers to an identity matrix and that v1,...,vq is a Hilbert basis. It is shown that if C has a perfect matching such that C has the packing property and g=2, then A diagonaliz…
The shuffle conjecture expresses a relationship between parking functions, diagonal harmonics, and the Bergeron-Garsia $\nabla$ operator. Recent conjectures about a family of modified Hall-Littlewood operators made by Haglund, Morse, and Zabrocki sharpen the shuffle conjecture and suggest a variety of combinatorial properties of parking functions. In particular, their conjectures combined with previously established commutativity laws of the Hall-Littlewood operators, suggest the existence of certain bijections relating parking functions with different diagonal compositions. In this paper we f…
We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case $a\in M_{2\times q}(2\mathbb N)$ to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of the "best algebraic normalization" of $I(a)$, in the case $a\in M_{2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a…
