128 research outputs found
Schur times Schubert via the Fomin-Kirillov algebra
We study multiplication of any Schubert polynomial by a
Schur polynomial (the Schubert polynomial of a Grassmannian
permutation) and the expansion of this product in the ring of Schubert
polynomials. We derive explicit nonnegative combinatorial expressions for the
expansion coefficients for certain special partitions , including
hooks and the 2x2 box. We also prove combinatorially the existence of such
nonnegative expansion when the Young diagram of is a hook plus a box
at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the
Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the
nonnegativity conjecture of Fomin and Kirillov.
This approach works in the more general setup of the (small) quantum
cohomology ring of the complex flag manifold and the corresponding (3-point)
Gromov-Witten invariants. We provide an algebro-combinatorial proof of the
nonnegativity of the Gromov-Witten invariants in these cases, and present
combinatorial expressions for these coefficients
Combinatorial description of the cohomology of the affine flag variety
International audienceWe construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety
On the commutative quotient of Fomin-Kirillov algebras
The Fomin-Kirillov algebra is a noncommutative algebra with a
generator for each edge in the complete graph on vertices. For any graph
on vertices, let be the subalgebra of
generated by the edges in . We show that the commutative quotient of
is isomorphic to the Orlik-Terao algebra of . As a
consequence, the Hilbert series of this quotient is given by , where is the chromatic polynomial of . We also
give a reduction algorithm for the graded components of that do
not vanish in the commutative quotient and show that their structure is
described by the combinatorics of noncrossing forests.Comment: 11 pages, 3 figure
PBW deformations of a Fomin-Kirillov algebra and other examples
We begin the study of PBW deformations of graded algebras relevant to the
theory of Hopf algebras. One of our examples is the Fomin-Kirillov algebra FK3.
Another one appeared in a paper of Garc\'ia Iglesias and Vay. As a consequence
of our methods, we determine when the deformations are semisimple and we are
able to produce PBW bases and polynomial identities for these deformations.Comment: 22 pages. Accepted for publication in Algebr. Represent. Theor
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