Related papers: Double Schubert polynomials do have saturated Newt…
Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide…
All Cohen--Macaulay polymatroidal ideals are classified. The Cohen--Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the squarefree Veronese ideals.
We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of…
We show that every convergent power series with monomial extended Jacobian ideal is right equivalent to a Thom-Sebastiani polynomial. This solves a problem posed by Hauser and Schicho. On the combinatorial side, we introduce a notion of…
We give a combinatorial proof that Postnikov and Stanley's formula for dual Schubert polynomials in terms of weighted chains in Bruhat order is equivalent to a classical Cauchy identity for polynomials. This gives a natural interpretation…
In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…
We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types B, C, D which naturally extends the family of Lascoux of Schutzenberger in type A. These polynomials satisfy positivity, orthogonality, and stability properties,…
We introduce the cave polynomial of a polymatroid and show that it yields a valuative function on polymatroids. The support of this polynomial after homogenization is again a polymatroid. The cave polynomial gives a $K$-theoretic…
We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.
Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric…
The graded cellularity of Libedinsky Double Leaves, which form a basis for the endomorphism ring of the Bott_Samelson_Soergel bimodules, allows us to view the Kazhdan_Lusztig polynomials as graded decomposition numbers. Using this point of…
In this work, we extend Saia's results on the characterization of Newton non-degenerate ideals to the context of ideals in $O_{X(S)}$, where $X(S)$ is an affine toric variety defined by the semigroup $S\subset \mathbb{Z}^{n}_{+}$. We…
We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also…
D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a…
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.…
Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers…
Schur polynomials are special cases of Schubert polynomials, which in turn are special cases of dual characters of flagged Weyl modules. The principal specialization of Schur and Schubert polynomials has a long history, with Macdonald…
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…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
We prove a formula for the multidegrees of a rational map defined by generalized monomials on a projective variety, in terms of integrals over an associated Newton region. This formula leads to an expression of the multidegrees as volumes…