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1. From Quasi-Symmetric to Schur Expansions with Applications to Symmetric Chain Decompositions and Plethysm

   https://doi.org/10.37236/12950  

   Orellana, Rosa; Saliola, Franco; Schilling, Anne; Zabrocki, Mike (October 2024, The Electronic Journal of Combinatorics)

   It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $$s_w[s_h](x,y)$$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $$w\times h$$ box. For $w=4$, this is the first known combinatorial expression for the coefficient of $$s_{\lambda}$$ in $$s_{w}[s_{h}]$$ for two-row partitions $$\lambda$$, and for $w=3$ the combinatorial expression is new. 

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2. Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture

   https://doi.org/10.1093/imrn/rnae090  

   Gillespie, Maria; Griffin, Sean T (May 2024, International Mathematics Research Notices)

   We prove that $$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $$\Delta $$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $$\Delta $$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $$t$$ and $$t^{2}$$ coefficients of $$\omega \Delta ^{\prime}_{e_{k}}e_{n}$$. 

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3. A cocharge formula for the ∆-Springer modules

   Gillespie, Maria; Griffin, Sean T (April 2023, Séminaire Lotharingien de Combinatoire)

   We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Sn-modules Rn,λ,s, which appear as the cohomology rings of the “∆-Springer” varieties. These modules interpolate between the Garsia-Procesi modules Rµ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture. Our formula directly generalizes the known cocharge formula for Garsia-Procesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing battery-powered tableaux that “store” extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and s ≤ ℓ(λ)+2, as well as for n ≤ 8 and s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s. 

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4. Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

   https://doi.org/10.1093/imrn/rnz261  

   Billey, Sara C.; Rhoades, Brendon; Tewari, Vasu (November 2019, International Mathematics Research Notices)

   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. 

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5. Almost isotropic Kähler manifolds

   https://doi.org/10.1515/crelle-2019-0030  

   Schmidt, Benjamin; Shankar, Krishnan; Spatzier, Ralf (October 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} . 

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