We study the multiplicity $b_S(n)$ of the trivial representation in the symmetric group representations $β_S$ on the (top) homology of the rank-selected partition lattice $Π_n^S$. We break the possible rank sets $S$ into three cases: (1) $1\not\in S$, (2) $S=1,..., i$ for $i\ge 1$ and (3) $S=1,..., i,j_1,..., j_l$ for $i,l\ge 1$, $j_1 > i+1$. It was previously shown by Hanlon that $b_S(n)=0$ for $S=1,..., i$. We use a partitioning for $Δ(Π_n)/S_n$ due to Hersh to confirm a conjecture of Sundaram that $b_S(n)>0$ for $1\not\in S$. On the other hand, we use the spectral sequence of a filtered complex to show $b_S(n)=0$ for $S=1,..., i,j_1,..., j_l$ unless a certain type of chain of support $S$ exists. The partitioning for $Δ(Π_n)/S_n$ allows us then to show that a large class of rank sets $S=1,..., i,j_1,..., j_l$ for which such a chain exists do satisfy $b_S(n)>0$. We also generalize the partitioning for $Δ(Π_n)/S_n$ to $Δ(Π_n)/S_λ$; when $λ= (n-1,1)$, this partitioning leads to a proof of a conjecture of Sundaram about $S_1\times S_{n-1}$-representations on the homology of the partition lattice.
We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.
The rational homology group of the order complex of non-even partitions of a finite set is calculated. A twisted version of the Goresky-MacPherson approach to similar homology calculations is proposed.
For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs in \cite{BW} using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$.
We determine all the multiplicity-free representations of the symmetric group. This project is motivated by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.
We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual operad. On the other hand, we get new methods for proving that an operad is Koszul.
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρ_{R_i} in the homology of the proper part \bar{L…
Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characteri…
This article, written in fond memory of Herbert Saul Wilf (June 13, 1931- Jan. 7, 2012), explores integer partitions where each part shows up a different number of times than the other parts (if it shows up at least once), thereby making a modest contribution towards the solution of one of the eight intrigiung problems posted by Herb Wilf on his website on Dec. 13, 2010
Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperpl…
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
The $q$-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostant's partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $\mathfrak{sl}_4(\mathbb{C})$ and give closed formulas for the $q$-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostant'…
Let $λ$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(λ)$ and $M_n=M_n(λ)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n -L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.
If $[λ(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $μ$ is a partition of $n$, we study the number $K_{[λ(j)]μ}$ of sequences of semistandard Young tableaux of shape $[λ(j)]$ and total weight $μ$. We show that the numbers $K_{[λ(j)] μ}$ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of conditions on $[λ(j)]$ and $μ$ which are equivalent to $K_{[λ(j)] μ} = 1$, generalizing a theorem of Berenshte\uın and Zelevinski\uı. We also show that the questions of whether $…
We study the asymptotic behavior of the maximal multiplicity $M_n=M_n(σ)$ of the blocks in a set partition of $[n]=\{1,2,...,n\}$, assuming that $σ$ is chosen uniformly at random from the set of all such partitions. Let $W=W(n)$ be the unique positive root of the equation $We^W=n$ and let $f_n$ be the fractional part of $W(n)$. Furthermore, let $R_n=W^{\lfloor W\rfloor}/\lfloor W\rfloor !$ and let $\vartheta_n=\min{\{f_n,1-f_n\}}$. We show that, over a subsequence $\{n_k\}_{k\ge 1}$, $(M_{n_k}-R_{n_k})/\sqrt{R_{n_k}}$ converges weakly, as $k\to\infty$, to $\max{\{Z_1,Z_2-u\}}$, where $Z_1$ and…
We consider the poset of weighted partitions $Π_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $Π_n^w$ provide a generalization of the lattice $Π_n$ of partitions, which we show possesses many of the well-known properties of $Π_n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and …
We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplect…
We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of $GL_n(\mathbb{C})$ indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of $S_n$ in an irreducible polynomial representation indexed by a hook partition.
For fixed weights w_1,...,w_n, and for d>0, we let B denote a collection of d*n balls, with d balls of weight w_i for each i=1,...,n. We consider the problem of assigning the balls to n bins with capacities C_1,...,C_n, in such a way that each bin is assigned d balls, without exceeding its capacity. When d>>0, we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w_i are distinct. The feasibility criteria that we present here are …
Given a filter $Δ$ in the poset of compositions of $n$, we form the filter $Π^{*}_Δ$ in the partition lattice. We determine all the reduced homology groups of the order complex of $Π^{*}_Δ$ as ${\mathfrak S}_{n-1}$-modules in terms of the reduced homology groups of the simplicial complex $Δ$ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank--Hanlon--Robinson and Wachs on the $d$-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to intege…
We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology …
We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's Theorem. If the alternating sum and the number of…
For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition $\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^{B}$ where $B$ is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homol…
We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs that compute for these Lie algebras: the multiplicity of a weight in an irreducible finite-dimensional representation; the decomposition coefficients of the tensor product of two irreducible finite-dimensional representations. These programs can also calculate associated Ehrhart quasipolynomials.
Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $ξ$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(ξ)=\sum c_i q^i$ where $c_i$ is the number of ways $ξ$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight represen…
It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions λ,μare big enough then the coefficients of the Kronecker product [λ][μ]=\sum_\n g(ł,\m,\n)[ν] do not depend on the first part but only on the other parts. The reduced Kronecker product [λ]_\bullet \star[μ]_\bullet can be viewed (roughly) as the Kronecker product [(n-|λ|,λ)][(n-|μ|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free a…
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-va…
We apply some recent developments of Baldoni-DeLoera-Vergne on vector partition functions, to Kostant and Steinberg formulas, in the case of $A_r$. We therefore get a fast {\sc Maple} program that computes for $A_r$: the multiplicity $c_{λ,μ}$ of the weight $μ$ in the representation $V(λ)$ of highest weight $λ$; the multiplicity $c_{λ,μ,ν}$ of the representation $V(ν)$ in $V(λ)\otimes V(μ)$. The computation also gives the locally polynomial functions $c_{λ,μ}$ and $c_{λ,μ,ν}$.
Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear ho…
This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tan…
This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $\mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-representation on the $i(d-1)^{st}$ cohomology stabilizes sharply at $n=3i$ (resp. $n=3i+1$) when $d$ is odd (resp. even). The result comes from analyzing $S_n$-representations known to control the cohomology: the Whitney homology of set partition lattices for $d$ even, and the h…
Let V be a symplectic vector space of dimension 2n. Given a partition λwith at most n parts, there is an associated irreducible representation S_{[λ]}(V) of Sp(V). This representation admits a resolution by a natural complex L^λ, which we call the Littlewood complex, whose terms are restrictions of representations of GL(V). When λhas more than n parts, the representation S_{[λ]}(V) is not defined, but the Littlewood complex L^λstill makes sense. The purpose of this paper is to compute its homology. We find that either L^λis acyclic or that it has a unique non-zero homology group, which forms a…
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs, whose blocks are of size $k$ taken from an $n$-set, is a partition of all the $k$-subsets of the $n$-set into disjoint copies of block designs, defined on the $n$-set, and with the same parameters. The current most intriguing question in this direction is whether large sets of Steiner quadruple systems exist and to provide explicit constructions for those para…
In this paper we apply a method of Robinson and Taulbee for computing Kronecker coefficients together with other ingredients and show that the multiplicity of each component in a Kronecker square can be obtained from an evaluation of a certain polynomial, which depends only on the component and is computed combinatorially. This polynomial has as many variables as the set of isomorphism classes of connected skew diagrams of size at most the depth of the component. We present two applications. The first is a contribution to Saxl conjecture, which asserts that the Kronecker square of the staircas…
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
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 $λ$ (a partition of $n$) is the number of multiset tableaux of shape $λ$ 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.
