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Sheila Sundaram

Publications and source records attributed to Sheila Sundaram.

30 records · Page 2Linked to original sources
arXiv2021

The plethystic inverse of the odd Lie representations

The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations. We determine the plethystic inverse of the alternating sum of the odd Lie characteristics.

math.CO↗
arXiv2021

Prime power variations of higher $Lie_n$ modules

We define, for each subset $S$ of the set $\mathcal{P}$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. $Lie_n^\emptyset$ is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra, while $Lie_n^\mathcal{P}$ is the module $C\!onj_n$ of the conjugacy action of $S_n$ on $n$-cycles. For arbitrary $S$ the module $Lie_n^{S}$ interpolates between the representations $Lie_n$ and $C\!onj_n.$ We consider the symmetric and exterior powers of $Lie_n^S.$ These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher $Lie_n^S$ modules can be elegantly expressed as a multiplicity-free sum of power sums. In particular this establishes the Schur positivity of new classes of sums of power sums. More generally, for each nonempty subset $T$ of positive integers we define a sequence of symmetric functions $f_n^T$ of homogeneous degree $n.$ We show that the series $\sum_{λ, λ_i\in T} p_λ$ can be expressed as symmetrised powers of the functions $f_n^T$, analogous to the higher Lie modules first defined by Thrall. This in turn allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between $f_n^T$, the conjugacy action and the Lie representation.

math.RT↗
arXiv2021

The reflection representation in the homology of subword order

We investigate the homology representation of the symmetric group on rank-selected subposets of subword order. We show that the homology module for words of bounded length, over an alphabet of size $n,$ decomposes into a sum of tensor powers of the $S_n$-irreducible $S_{(n-1,1)}$ indexed by the partition $(n-1,1),$ recovering, as a special case, a theorem of Björner and Stanley for words of length at most $k.$ For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)}$, and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted. We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}$, and show that its Frobenius characteristic is $h$-positive and supported on the set $T_{1}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 1\}.$ Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of the Schur function $s_{(n-1,1)},$ as an integer combination of the set $T_{2}(n)=\{h_λ: λ=(n-r, 1^r), r\ge 2\}.$ We conjecture that this combination is nonnegative, establishing this fact for particular cases.

math.RT↗
arXiv2021

Vector partition functions and Kronecker coefficients

The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group $GL(n m)$ into irreducibles for the subgroup $GL(n)\times GL(m)$. In this work we study the quasipolynomial nature of the Kronecker function using elementary tools from polyhedral geometry. We write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker coefficients. Our derivation is explicit and self-contained, and gives a new exact formula and an upper bound for the Kronecker coefficients in the first nontrivial case.

math.RT↗
arXiv2020

On a curious variant of the $S_n$-module $Lie_n$

We introduce a variant of the much-studied $Lie$ representation of the symmetric group $S_n$, which we denote by $Lie_n^{(2)}.$ Our variant gives rise to a decomposition of the regular representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincaré-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised $Lie$ modules. We show that nearly every known property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.

math.RT↗
arXiv2020

Variations on the $S_n$-module $Lie_n$

We define, for each subset $S$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. When $S=\emptyset,$ this is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra. The most intriguing case is $S=\{2\},$ giving a decomposition of the regular representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincaré-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised $Lie$ modules. We show that nearly every known property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh. For arbitrary $S,$ the symmetric and exterior powers of the module $Lie_n^S$ allow us to deduce Schur positivity for a new class of multiplicity-free sums of power sums.

math.RT↗
arXiv2018

On a positivity conjecture in the character table of $S_n$

In previous work of this author it was conjectured that the sum of power sums $p_λ,$ for partitions $λ$ ranging over an interval $[(1^n), μ]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $μ\in [(n-4,1^4), (n)]$ or $μ\in [(1^n), (3,1^{n-3})], $ or $μ=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.

math.CO↗
arXiv2018

On conjugacy classes of $S_n$ containing all irreducibles

It is shown that for the conjugation action of the symmetric group $S_n,$ when $n=6$ or $n\geq 8,$ all $S_n$-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition $λ$ of $n$ with at least two parts, whose parts are all distinct and taken from the set of odd primes and 1. The following simple characterisation of conjugacy classes containing all irreducibles is proved: If $n\neq 4,8,$ the partition $λ$ of $n$ indexes a global conjugacy class for $S_n$ if and only if it has at least two parts, and all its parts are odd and distinct.

math.GR↗
arXiv2017

The conjugacy action of $S_n$ and modules induced from centralisers

We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power-sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension $n!$ with this property, including a twisted analogue of the conjugacy action.

math.RT↗
arXiv2005

Counting permutations by congruence class of major index

Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i (mod k) and maj pi^{-1} = j (mod l). We prove bijectively that if k,l are relatively prime and at most n then m_n^{k,l}(i,j) = n!/(kl) which, surprisingly, does not depend on i and j. Equivalently, if m_n^{k,l}(i,j) is interpreted as the (i,j)-entry of a matrix m_n^{k,l}, then this is a constant matrix under the stated conditions. This bijection is extended to show the more general result that for d at least 1 and k,l relatively prime, the matrix m_n^{kd,ld} admits a block decompostion where each block is the matrix m_n^{d,d}/(kl). We also give an explicit formula for m_n^{n,n} and show that if p is prime then m_{np}^{p,p} has a simple block decomposition. To prove these results, we use the representation theory of the symmetric group and certain restricted shuffles.

math.CO↗
arXiv2001

On counting permutations by pairs of congruence classes of major index

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).

math.CO↗