Alexander Yong
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View article: Newell–Littlewood numbers III: Eigencones and GIT-semigroups
Newell–Littlewood numbers III: Eigencones and GIT-semigroups Open
The Newell–Littlewood (NL) numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood–Richardson (LR) coefficients form a special case. Klyachko connected eigenvalues of sums of Hermi…
View article: RSK as a linear operator
RSK as a linear operator Open
The Robinson-Schensted-Knuth correspondence (RSK) is a bijection between nonnegative integer matrices and pairs of Young tableaux. We study it as a linear operator on the coordinate ring of matrices, proving results about its diagonalizabi…
View article: Representations from matrix varieties, and filtered RSK
Representations from matrix varieties, and filtered RSK Open
Matrix Schubert varieties (Fulton '92) carry natural actions of Levi groups. Their coordinate rings are thereby Levi-representations; what is a combinatorial counting rule for the multiplicities of their irreducibles? When the Levi group i…
View article: The Kostka semigroup and its Hilbert basis
The Kostka semigroup and its Hilbert basis Open
The Kostka semigroup consists of pairs of partitions with at most r parts that have positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, thr…
View article: Combinatorial commutative algebra rules
Combinatorial commutative algebra rules Open
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and re…
View article: Schubert determinantal ideals are Hilbertian
Schubert determinantal ideals are Hilbertian Open
Abhyankar defined an ideal to be Hilbertian if its Hilbert polynomial coincides with its Hilbert function for all nonnegative integers. In 1984, he proved that the ideal of (r+1)-order minors of a generic p x q matrix is Hilbertian. We giv…
View article: Levi-spherical Schubert varieties
Levi-spherical Schubert varieties Open
We prove a short, root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety $G/B$ of finite Lie type. We apply this to the study of multiplicity-free decompositions of a Demazur…
View article: Schubert geometry and combinatorics
Schubert geometry and combinatorics Open
This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.
View article: Newell–Littlewood numbers II: extended Horn inequalities
Newell–Littlewood numbers II: extended Horn inequalities Open
The Newell–Littlewood numbers N μ,ν,λ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions (μ,ν,λ) does N μ,ν,λ >0 hold? The Littlewood–Richardson coefficient case…
View article: Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture
Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture Open
A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of…
View article: Reduced Word Enumeration, Complexity, and Randomization
Reduced Word Enumeration, Complexity, and Randomization Open
A reduced word of a permutation w is a minimal length expression of w as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that …
View article: Castelnuovo-Mumford regularity and Schubert geometry
Castelnuovo-Mumford regularity and Schubert geometry Open
We study the Castelnuovo-Mumford regularity of tangent cones of Schubert varieties. Conjectures about this statistic are presented; these are proved for the covexillary case. This builds on work of L. Li and the author on these tangent con…
View article: Minimal equations for matrix Schubert varieties
Minimal equations for matrix Schubert varieties Open
Explicit minimal generators for Fulton's Schubert determinantal ideals are determined along with some implications.
View article: Proper permutations, Schubert geometry, and randomness
Proper permutations, Schubert geometry, and randomness Open
We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.
View article: Newell-Littlewood numbers III: eigencones and GIT-semigroups
Newell-Littlewood numbers III: eigencones and GIT-semigroups Open
The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matri…
View article: Classification of Levi-spherical Schubert varieties
Classification of Levi-spherical Schubert varieties Open
A Schubert variety in the complete flag manifold $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has a dense orbit. We give a combinatorial classification of these Schubert varieties.…
View article: An efficient algorithm for deciding vanishing of Schubert polynomial coefficients
An efficient algorithm for deciding vanishing of Schubert polynomial coefficients Open
Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau cr…
View article: The "Grothendieck to Lascoux" conjecture
The "Grothendieck to Lascoux" conjecture Open
This report formulates a conjectural combinatorial rule that positively expands Grothendieck polynomials into Lascoux polynomials. It generalizes one such formula expanding Schubert polynomials into key polynomials, and refines another one…
View article: The Kostka semigroup and its Hilbert basis
The Kostka semigroup and its Hilbert basis Open
The Kostka semigroup consists of pairs of partitions with at most r parts that have positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, thr…
View article: Newell-Littlewood numbers
Newell-Littlewood numbers Open
The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Ko…
View article: Generalized permutahedra and Schubert calculus
Generalized permutahedra and Schubert calculus Open
We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which …
View article: Newell-Littlewood numbers II: extended Horn inequalities
Newell-Littlewood numbers II: extended Horn inequalities Open
The Newell-Littlewood numbers $N_{μ,ν,λ}$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $(μ,ν,λ)$ does $N_{μ,ν,λ}>0$ hold? The Littlewood-Richardson coeffic…
View article: Coxeter combinatorics and spherical Schubert geometry
Coxeter combinatorics and spherical Schubert geometry Open
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
View article: Multiplicity-free key polynomials
Multiplicity-free key polynomials Open
The key polynomials, defined by A. Lascoux-M.-P. Schützenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorial models for key polynomials. The first is du…
View article: Newell-Littlewood numbers
Newell-Littlewood numbers Open
The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Ko…
View article: The Prism tableau model for Schubert polynomials
The Prism tableau model for Schubert polynomials Open
The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly i…
View article: Equivariant cohomology, Schubert calculus, and edge labeled tableaux
Equivariant cohomology, Schubert calculus, and edge labeled tableaux Open
This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences f…
View article: The A.B.C.Ds of Schubert calculus
The A.B.C.Ds of Schubert calculus Open
We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work…
View article: An estimation method for game complexity
An estimation method for game complexity Open
We looked at a method for estimating the complexity measure of game tree size (the number of legal games). It seems effective for a number of children's games such as Tic-Tac-Toe, Connect Four and Othello.