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Abstract Let$${\mathbf {x}}_{n \times n}$$be an$$n \times n$$matrix of variables, and let$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$be the polynomial ring in these variables over a field$${\mathbb {F}}$$. We study the ideal$$I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$is the generating function of permutations in$${\mathfrak {S}}_n$$by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space ofk-local permutation statistics. We also calculate the structure of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$as a graded$${\mathfrak {S}}_n \times {\mathfrak {S}}_n$$-module. 

Abstract Let$$\Omega _n$$be the ring of polynomial-valued holomorphic differential forms on complexn-space, referred to in physics as the superspace ring of rankn. The symmetric group$${\mathfrak {S}}_n$$acts diagonally on$$\Omega _n$$by permuting commuting and anticommuting generators simultaneously. We let$$SI_n \subseteq \Omega _n$$be the ideal generated by$${\mathfrak {S}}_n$$-invariants with vanishing constant term and study the quotient$$SR_n = \Omega _n / SI_n$$of superspace by this ideal. We calculate the doubly-graded Hilbert series of$$SR_n$$and prove an ‘operator theorem’, which characterizes the harmonic space$$SH_n \subseteq \Omega _n$$attached to$$SR_n$$in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki. 

Abstract We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $$G_{Q,\gamma }$$ constructed from a symmetric quiver $$Q$$ with enough loops and a dimension vector $$\gamma $$. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of $$S_{\gamma }$$-invariants of the Postnikov–Shapiro slim subgraph space attached to $$G_{Q,\gamma }$$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of $$S_{\gamma }$$-orbits under the permutation action on the set of break divisors on $$G$$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest. 

Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory.