Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

Abstract

Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial$B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

Root Cones and the Resonance Arrangement

“…Dually, the collection of unbalanced families is closed under taking intersections, which inspired the investigation of maximal unbalanced families. In the work of Billera, Tatch Moore, Dufort Moraites, Wang, and Williams [7], it is recognized that maximal unbalanced families are in bijection with chambers of a hyperplane arrangement which we refer to as the resonance arrangement, following [6,9].…”

“… Kamiya, Takemura, and Terao call the resonance arrangement the all-subsets arrangement, and that name is also used by Billera, Tatch Moore, Dufort Moraites, Wang, and Williams. We adopt the nomenclature of Cavalieri et al which is also followed in later work on Hurwitz numbers and is used in recent work of Billera, Billey, and Tewari[6]. In Liu, Norledge, and Ocneanu[28], the resonance arrangement is also called the adjoint braid arrangement.the electronic journal of combinatorics 28(1) (2021), #P1 12.…”

Root Cones and the Resonance Arrangement

“…The Kostant partition function (for the root system A n ) is a counting function κ n : R n+1 → Z ≥0 . For a given point a ∈ R n+1 , we have (6) κ n (a) = x ∈ Z ( n+1 2 )…”

“…1 Kamiya, Takemura, and Terao call the resonance arrangement the all-subsets arrangement, and that name is also used by Billera, Tatch Moore, Dufort Moraites, Wang, and Williams. We adopt the nomenclature of Cavalieri et al which is also followed in later work on Hurwitz numbers and is used in recent work of Billera, Billey, and Tewari [6]. In Liu, Norledge, and Ocneanu [25], the resonance arrangement is also called the adjoint braid arrangement.…”

Hopf monoids, permutohedral cones, and generalized retarded functions

“…This hyperplane arrangement has several names. It is known as the restricted all-subset arrangement [KTT11], [KTT12], [BMM + 12], the resonance arrangement [CJM11], [Cav16], [BBT18], [GMP19], and the root arrangement [LMPS19]. Its spherical representation is called the Steinmann planet by physicists [BG67], [Eps16].…”