For a simplicial complex $\Delta$ and integers $n_i\ge 2$, set $A_{\Delta,\mathbf n}=\mathbb{k}[x_1,\ldots,x_m]/(I_\Delta+(x_1^{n_1},\ldots,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant ghost vertices. The formula extends the free-face formula of Gasanova--Herzog--Hibi--Moradi for square-zero flag face algebras and recovers, in the simplex-boundary case, a special case of their formula for monomial almost complete intersections. As a first consequence, we classify the nearly Gorenstein algebras in this family: on each connected component $C$ of $\Delta^{(1)}$, the induced complex is either the simplex $2^C$, with arbitrary exponents, or the boundary $\partial 2^C$, with every exponent equal to two. We also compute the Teter number on this nearly Gorenstein locus. For flag complexes the trace is generated by the free-face monomials for arbitrary exponents, and we characterize the equalities $\operatorname{tr}_A(\omega_A)=\mathfrak m_A^q$. In the square-zero one-dimensional case we isolate the additional contribution coming from triangle components.
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0<k<\min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood--Richardson tableaux.
We study the Schur-positive cone in $\Rspace_{n,\mathbb R}\coloneqq \operatorname{span}_{\mathbb R}\{p_d^{n/d}:d\mid n\}$ through its basis $Q_{n,d}\coloneqq\ell_{n/d}^{(1)}[p_d]$, where $\ell_m^{(1)}$ is the Frobenius characteristic of the representation induced to $S_m$ from a faithful linear character of the subgroup generated by an $m$-cycle; brackets denote plethysm. A Boolean $Q$-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean $Q$-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set $J$ of divisors of $n$, the sum $\sum_{d\in J}Q_{n,d}$ is Schur-positive exactly when $1\in J$ and, for even $n$, $n\in J$ implies $n/2\in J$. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function $\sum_{d\mid n}c_d(n/d)^2p_d^{n/d}$, where $c_d(r)$ is the Ramanujan sum, has a positive coefficient of $s_\lambda$ for every $n\ge1$ and $\lambda\vdash n$, except when $n\equiv2\pmod4$ and $\lambda=(1^n)$, in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its $Q$-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with $Q_{n,1}$-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For $n\ge18$, we prove that setting the coefficient of $s_{(n)}$ or $s_{(1^n)}$ equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with $Q_{n,1}$-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.
Let $X_{G_h}(\mathbf{x};q)=\sum_{\mu\vdash n}c_\mu(q)e_\mu(\mathbf{x})$ be the chromatic quasisymmetric function of the natural unit interval graph attached to a Hessenberg function $h$. We establish an infinite class, valid in all orders, for which every nonzero polynomial $c_\mu(q)$ has a nonnegative, log-concave coefficient sequence with interval support. Namely, this holds whenever $h$ is abelian and its complement-Ferrers partition $\lambda$ satisfies $\min\{\lambda_1,\ell(\lambda)\}\leq 3$; equivalently, the diagram has at most three rows or at most three columns. Cubic interpolation reduces the rank-three case to a uniform theorem for a difference of two products of four $q$-integers, proved by positive decomposition, interval methods, and finite-window smoothing. The argument also yields explicit formulas for every supported elementary coefficient in complement-Ferrers rank at most three. We also include a connected 13-vertex natural unit interval graph for which one elementary coefficient is positive, palindromic, and unimodal but not log-concave, thereby recording the failure of the unrestricted conjecture. Thus low complement-Ferrers rank gives a substantial positive regime even though coefficientwise $e$-log-concavity fails in general.
For a finite connected simplicial complex $X$, the strand map $\iota_\ast$, from $\mathbb{P}_n(X)$ to $\prod_{i=1}^n\pi_1(X,x_i^0)$, sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than $S^2$ and $\mathbb{RP}^2$: the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\textit{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\ker\iota_\ast=\left\langle \operatorname{im}(\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)) \right\rangle$, and $\textit{Goldberg}$ if $X_0$ can moreover be chosen so that $\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)$ is injective. We prove that the strand map is surjective if and only if $X\not\cong S^1$; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\textit{admissible tree}$ -- a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces $S^2$ and $\mathbb{RP}^2$, and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-modules. The key input is a mixed local reduction for finite Maurer--Cartan objects over the mixed de Rham--Dolbeault dga $(\wedge^\bullet V^\vee,d_V)$: a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact $X$, this gives an exact equivalence between the homotopy category of bounded finite-rank flat $V$-superconnections and $D^b_{\mathrm{coh}}(X,{\mathcal O}_V)$, interpolating between the de Rham and Dolbeault realizations. We prove locally finite \v{C}ech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on $S^1$ and $S^2$ delimit when ordinary monodromy $1$-groupoids can recover the derived category.
We show that, in the ring of virtual species, \[ \mathcal{I}=\sum_{m\geq 0}(-1)^m\prod_{i=1}^{m}\bigl((E^{-1})^i-1\bigr), \] where $\mathcal{I}$ is the species of interval orders and $E^{-1}$ is the multiplicative inverse of the species $E$ of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of $\mathcal{I}$ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity $\mathcal{I}=\mathcal{R}\circ E_+$, where $\mathcal{R}$ is the species of rigid interval orders. We also show that Glaisher's T-number $T_n$ counts the $24$-colored interval orders on $[n]$ in which no isolated element has color $24$.
Anders Claesson· 0 citations
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