We prove the sign-isotypic components of the coinvariant rings $R_n^{(2,1)}$ and $R_n^{(2,0)} \otimes R_n^{(0,1)}$ are isomorphic and show that the triply-graded multiplicity of this sign character is the Schr\"oder polynomial $S_n(q,t,a)$, divided by $1+a$. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of $R_n^{(2,1)}$. Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky--Mellit (2026) which relates the Khovanov--Rozansky homology of the $(n,n+1)$-torus knot to $R_n^{(2,0)} \otimes R_n^{(0,1)}$, by showing that the associated Poincar\'e series for this knot can be computed from the sign component of $R_n^{(2,1)}$.
For a field $F$ we study the coinvariants for the $SL_n(F)$-action on the double Steinberg module $St_n(F) \otimes St_n(F)$ and show they have a rich algebraic structure: for $n = 2$ it is the Grothendieck-Witt group of $F$, and for all $n$ they assemble to a graded nonunital $\mathbb{Z}[F^\times]$-algebra, whose ratio...
Tatiana Abdelnaim, David Chan, A. Kupers et al.· 0 citations
Let $P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. For each $p\in\{4,5,6,7,8\}$, we prove that the coefficients of $P_n(q)^p$ in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coeffici...
Let $(d_i(n))_{i=0}^n$ be the Boros--Moll coefficient sequence. We prove that, for every integer $n\ge1$, the polynomial \[ M_n(x)=\sum_{i=0}^n \bigl(d_i(n)^2-d_{i-1}(n)d_{i+1}(n)\bigr)x^i \] has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conj...
We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, and Brassil and Reichstein proved the conj...
We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer $n$, the polynomial \[ P_n(q) = \frac{(1-q)^2}{(1-q^{10n-1})(1-q^{15n-1})}\qbin{30n}{5n} \] has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem thr...
We show that an ir{\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O((\log_2 N)^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely iden...
Soumya Bhattacharya· 0 citations
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