arXiv subjects

Sheila Sundaram

Publications and source records attributed to Sheila Sundaram.

At least 19 recordsLinked to original sources

arXiv2026

Stability and ribbon bases for the rank-selected homology of geometric lattices

This paper analyzes the representation theoretic stability, in the sense of Thomas Church and Benson Farb, of the rank-selected homology of the Boolean lattice and the partition lattice, proving sharp uniform representation stability bounds in both cases. It proves a conjecture of the first author and Reiner by giving the sharp stability bound for general rank sets for the partition lattice. Along the way, a new homology basis sharing useful features with the polytabloid basis for Specht modules is introduced for the rank-selected homology and for the rank-selected Whitney homology of any geometric lattice, resolving an old open question of Bj\"orner. These bases give a matroid theoretic analogue of Specht modules.

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arXiv2025

Segre powers of posets preserve EL-shellability

For a bounded and graded poset $P$, we show that if $P$ is EL-shellable, then so is its $t$-fold Segre power $P^{(t)}=P\circ \cdots \circ P$ ($t$ factors), as defined by Bj\"orner and Welker [J. Pure Appl. Algebra, 198(1-3), 43--55 (2005)]. Our EL-labeling leads to formulas for the rank-selected invariants of $P^{(t)}$, generalising those given by Stanley for the subspace lattice [J. Combinatorial Theory Ser. A, 20(3):336-356, 1976].

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arXiv2025

Minimal elements in the skew extended 0-Hecke poset

The row-strict 0-Hecke action on standard immaculate skew tableaux was studied by the present authors, who showed that it gives rise to a bounded poset, called the \emph{skew immaculate Hecke poset}, and consequently to a cyclic 0-Hecke module. It was further shown that the subposet of skew standard extended immaculate tableaux always has a unique maximal element, but may have multiple minimal elements. In this paper we focus on these minimal elements, completely classifying them for a family of skew shapes that we call \emph{lobsters}. Moreover, we prove that when the skew shape is connected, the skew extended Hecke poset does have a unique minimal element, thereby showing that the associated 0-Hecke module is cyclic for both the row-strict and the dual immaculate actions.

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arXiv2025

Ordered set partition posets

A set partition is said to be ordered if the blocks of the partition are listed in a specific order. The ordered set partitions of $\{1,\ldots,n\}$, with a unique minimal element adjoined, form a lattice $\Om_n$ with respect to refinement. The lattice $\Om_n$ is well known to be the face lattice of the permutohedron. In this paper we study the combinatorics and topology of two subposets of $\Om_n$ with restricted block sizes, either all divisible by some fixed $d\ge2$, or all congruent to $1$ modulo $d$. For the $d$-divisible case we derive an explicit recursive atom ordering for the lattice, as well as formulas for the action of the symmetric group on the Whitney homology and the rank-selected homology, and also for the multiplicity of the trivial representation. In the 1 mod $d$ case we show that the poset has a curious interval structure related to the $k$-Catalan numbers. Our investigations lead to enumerative invariants in both cases. Open problems and avenues for future research are scattered throughout.

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arXiv2024

The skew immaculate Hecke poset and 0-Hecke modules

The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.

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arXiv2024

Homology of Segre powers of Boolean and subspace lattices

Segre products of posets were defined by Bj\"orner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric group $\sym_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into $\sym_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group $\sym_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the $\sym_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice.

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arXiv2024

Topology of Cut Complexes II

We continue the study of the $k$-cut complex $\Delta_k(G)$ of a graph $G$ initiated in the paper of Bayer, Denker, Jeli\'c Milutinovi\'c, Rowlands, Sundaram and Xue [Topology of cut complexes of graphs, SIAM J. on Discrete Math. 38(2): 1630--1675 (2024)]. We give explicit formulas for the $f$- and $h$-polynomials of the cut complex $\Delta_k(G_1+G_2) $ of the disjoint union of two graphs $G_1$ and $G_2$, and for the homology representation of $\Delta_k(K_m+K_n)$. We also study the cut complex of the squared path and the grid graph. Our techniques include tools from combinatorial topology, discrete Morse theory and equivariant poset topology.

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arXiv2024

The $e$-positivity of the chromatic symmetric function for twinned paths and cycles

The operation of twinning a graph at a vertex was introduced by Foley, Ho\`ang, and Merkel (2019), who conjectured that twinning preserves $e$-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that $e$-positivity is preserved by the twinning operation on cycles, by giving an $e$-positive generating function for the chromatic symmetric function, as well as an $e$-positive recurrence. We derive similar $e$-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities.

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arXiv2024

Koszulity, supersolvability, and Stirling representations

Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.

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arXiv2023

Moments of Colored Permutation Statistics on Conjugacy Classes

In this paper, we consider the moments of statistics on conjugacy classes of the colored permutation groups $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We first show that any fixed moment coincides on all conjugacy classes where all cycles have sufficiently long length. Additionally, for permutation statistics that can be realized via a process we call symmetric extensions, these moments are polynomials in $n$. Finally, for the descent statistic on the hyperoctahedral group $B_n\cong \mathfrak{S}_{n,2}$, we show that its distribution on conjugacy classes without short cycles satisfies a central limit theorem. Our results build on and generalize previous work of Fulman (\textit{J. Comb. Theory Ser. A.}, 1998), Hamaker and Rhoades (arXiv, 2022), and Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin (arXiv, 2023). In particular, our techniques utilize the combinatorial framework introduced by Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin.

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arXiv2023

Ramanujan sums and rectangular power sums

For a fixed nonnegative integer $u$ and positive integer $n$, we investigate the symmetric function \[\sum_{d|n} \left(c_d(\tfrac{n}{d})\right)^u p_d^{\tfrac{n}{d}},\] where $p_n$ denotes the $n$th power sum symmetric function, and $c_d(r)$ is a Ramanujan sum, equal to the sum of the $r$th powers of all the primitive $d$th roots of unity. We establish the Schur positivity of these functions for $u=0$ and $u=1$, showing that, in each case, the associated representation of the symmetric group $\mathfrak{S}_n$ decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case $u= 2$.

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arXiv2023

Permutation Statistics in Conjugacy Classes of the Symmetric Group

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of inversions, the number of descents, the major index, and the number of excedances. As a consequence, we obtain explicit formulas for the first moments of several statistics by conjugacy class. We also show that when the cycle lengths are sufficiently large, the higher moments of arbitrary permutation statistics are independent of the conjugacy class. Fulman (J. Comb. Theory Ser. A., 1998) previously established this result for major index and descents. We obtain these results, in part, by generalizing the techniques of Fulman (ibid.), and introducing the notion of permutation constraints. For permutation statistics that can be realized via symmetric constraints, we show that each moment is a polynomial in the degree of the symmetric group.

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arXiv2023

Topology of Cut Complexes of Graphs

We define the $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. This generalizes the Alexander dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner (1998). We describe the effect of various graph operations on the cut complex, and study its shellability, homotopy type and homology for various families of graphs, including trees, cycles, complete multipartite graphs, and the prism $K_n \times K_2$, using techniques from algebraic topology, discrete Morse theory and equivariant poset topology.

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arXiv2022

Pieri rules for skew dual immaculate functions

In this paper we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood-Richardson rule for Hopf algebras of Lam-Lauve-Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.

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arXiv2022

On Sums, Derivatives, and Flips of Riordan Arrays

We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the $A$- and $Z$-sequences of these sums of Riordan arrays, and also identify an analog for $A$-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations `Der' and `Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.

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arXiv2022

Total Cut Complexes of Graphs

Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex whose facets are the complements of independent sets of size $k$ in $G$. We study the homotopy types and combinatorial properties of total cut complexes for various families of graphs, including chordal graphs, cycles, bipartite graphs, the prism $K_n \times K_2$, and grid graphs, using techniques from algebraic topology and discrete Morse theory.

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arXiv2022

Row-strict dual immaculate functions

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We establish that this definition gives a function that can also be obtained by applying the $\psi$ involution to the dual immaculate functions of Berg, Bergeron, Saliola, Serrano, and Zabrocki (2014) and establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to Berg et. al (2014). We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions and establishing combinatorial properties for them.

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arXiv2022

0-Hecke modules for row-strict dual immaculate functions

We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\psi$ on the ring of quasisymmetric functions. We give an explicit description of the effect of $\psi$ on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing that \emph{all} the possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.

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