133 citations · 554 across the 117 of their papers we have counts for
27 papers · 1 filter
A quantum Murnaghan--Nakayama rule for the flag manifold
Carolina Benedetti, Nantel Bergeron, Laura Colmenarejo +2
In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate ste…
Two Murnaghan-Nakayama rules in Schubert calculus
Andrew Morrison, Frank Sottile
The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule…
A combinatorial proof that Schubert vs. Schur coefficients are nonnegative
Sami Assaf, Nantel Bergeron, Frank Sottile
We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf's theory of dual eq…
An inequality of Kostka numbers and Galois groups of Schubert problems
Christopher J. Brooks, Abraham Martin del Campo, Frank Sottile
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument…
Cofree compositions of coalgebras (extended abstract)
Stefan Forcey, Aaron Lauve, Frank Sottile
We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree…
Cofree compositions of coalgebras
Stefan Forcey, Aaron Lauve, Frank Sottile
We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofr…
Hopf structures on the multiplihedra
Stefan Forcey, Aaron Lauve, Frank Sottile
We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Mo…
New Hopf Structures on Binary Trees (Extended Abstract)
Stefan Forcey, Aaron Lauve, Frank Sottile
The multiplihedra {M_n} form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic com…
Skew Littlewood-Richardson rules from Hopf algebras
Thomas Lam, Aaron Lauve, Frank Sottile
We use Hopf algebras to prove a version of the Littlewood-Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlew…
A Littlewood-Richardson rule for Grassmannian Permutations
Kevin Purbhoo, Frank Sottile
We give a combinatorial rule for computing intersection numbers on a flag manifold which come from products of Schubert classes pulled back from Grassmannian projections. This rule…
The Horn recursion for Schur P- and Q- functions: Extended Abstract
Kevin Purbhoo, Frank Sottile
A consequence of work of Klyachko and of Knutson-Tao is the Horn recursion to determine when a Littlewood-Richardson coefficient is non-zero. Briefly, a Littlewood-Richardson coeff…
Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones
Matthias Beck, Christian Haase, Frank Sottile
We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expos…
Irrational proofs for three theorems of Stanley
Matthias Beck, Frank Sottile
We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley's reciprocity theorem, relates the rational gen…
Structure of the Loday-Ronco Hopf algebra of trees
Marcelo Aguiar, Frank Sottile
Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commut…
A Pieri-type formula for the K-theory of a flag manifold
Cristian Lenart, Frank Sottile
We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert cla…
Grothendieck polynomials via permutation patterns and chains in the Bruhat order
Cristian Lenart, Shawn Robinson, Frank Sottile
We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear com…
Quiver coefficients are Schubert structure constants
Anders Skovsted Buch, Frank Sottile, Alexander Yong
We give an explicit natural identification between the quiver coefficients of Buch and Fulton, decomposition coefficients for Schubert polynomials, and the Schubert structure const…
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
Marcelo Aguiar, Nantel Bergeron, Frank Sottile
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field equipped with a character (multiplicative linear functional) . We show that the terminal…
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)
Marcelo Aguiar, Frank Sottile
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determ…
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations
Marcelo Aguiar, Frank Sottile
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determ…
Skew Schubert polynomials
Cristian Lenart, Frank Sottile
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a r…
Complementary Algorithms For Tableaux
Tom Roby, Frank Sottile, Jeffrey Stroomer +1
We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fou…
Non-commutative Pieri operators on posets
Nantel Bergeron, Stefan Mykytiuk, Frank Sottile +1
We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representatio…
Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions
Nantel Bergeron, Stefan Mykytiuk, Frank Sottile +1
In his work on P-partitions, Stembridge defined the algebra of peak functions Pi, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show t…