Colleen Robichaux
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View article: Vanishing of Schubert coefficients in probabilistic polynomial time
Vanishing of Schubert coefficients in probabilistic polynomial time Open
The Schubert vanishing problem asks whether Schubert structure constants are zero. We give a complete solution of the problem from an algorithmic point of view, by showing that Schubert vanishing can be decided in probabilistic polynomial …
View article: On Vanishing of Gromov--Witten Invariants
On Vanishing of Gromov--Witten Invariants Open
We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the …
View article: Signed combinatorial interpretations in algebraic combinatorics
Signed combinatorial interpretations in algebraic combinatorics Open
We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert theor…
View article: Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz
Vanishing of Schubert coefficients via the effective Hilbert nullstellensatz Open
Schubert Vanishing is a problem of deciding whether Schubert coefficients are zero. Until this work it was open whether this problem is in the polynomial hierarchy ${{\mathsf {PH}}}$ . We prove this problem is in ${{\mathsf {AM}}} \cap {{\…
View article: Signed combinatorial interpretations in algebraic combinatorics
Signed combinatorial interpretations in algebraic combinatorics Open
We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert theor…
View article: A counterexample to the Ross--Yong conjecture for Grothendieck polynomials
A counterexample to the Ross--Yong conjecture for Grothendieck polynomials Open
We give a minimal counterexample for a conjecture of Ross and Yong (2015) which proposes a K-Kohnert rule for Grothendieck polynomials. We conjecture a revised version of this rule. We then prove both rules hold in the $321$-avoiding case.
View article: Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties
Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties Open
We prove the Castelnuovo--Mumford regularity of 321-avoiding Kazhdan--Lusztig varieties can be computed combinatorially in terms of $K$-theoretic skew excited Young diagrams. We present an algorithm which gives a lower bound for this regul…
View article: Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties
Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties Open
We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which …
View article: Shifted edge labeled tableaux and localizations
Shifted edge labeled tableaux and localizations Open
We prove cases of a conjectural rule of H. Yadav, A. Yong, and the author for structure coefficients of the D. Anderson-W. Fulton ring. In particular, we give a combinatorial description for certain localization coefficients of this ring, …
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: Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity
Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity Open
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation.…
View article: Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford\n regularity
Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford\n regularity Open
We give an explicit formula for the degree of the Grothendieck polynomial of\na Grassmannian permutation and a closely related formula for the\nCastelnuovo-Mumford regularity of the Schubert determinantal ideal of a\nGrassmannian permutati…
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: Computational complexity, Newton polytopes, and Schubert polynomials
Computational complexity, Newton polytopes, and Schubert polynomials Open
The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby…
View article: Complexity, combinatorial positivity, and Newton polytopes.
Complexity, combinatorial positivity, and Newton polytopes. Open
The Nonvanishing Problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby…
View article: Vanishing of Littlewood-Richardson polynomials is in P
Vanishing of Littlewood-Richardson polynomials is in P Open
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert c…
View article: K-Knuth Equivalence for Increasing Tableaux
K-Knuth Equivalence for Increasing Tableaux Open
A $K$-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced by Buch, Kresch, Shimozono, Tamvakis, and Yong (2006) and Buch and Samuel (2013), respectively. The resulting $K$-Knuth equivalence relations on …