Multiplication formulas in the K-theory of complex flag varieties

Cristian Lenart

SUNY Albany

May 10,
Special Time: **3:00pm**
2-338

ABSTRACT 

The main object of the talk is to present some explicit formulas for expanding the product of certain Grothendieck polynomials in the basis of such polynomials. Grothendieck polynomials are representatives for Schubert classes in the K-theory of the variety of complete flags in the complex $n$-dimensional space; thus, they generalize Schubert polynomials, which are representatives for Schubert classes in cohomology. Our formulas are concerned with the multiplication of an arbitrary Grothendieck polynomial by one indexed by a simple transposition, and, more generally, by a cycle of the form $(i,i+1,...,i+p)$; in other words, we generalize Monk's and Pieri's formulas for Schubert polynomials. Our formulas are in terms of chains in a suborder of the Bruhat order on the symmetric group (known as k-Bruhat order) with certain labels on its covers. We deduce as corollaries A. Lascoux's transition formula for Grothendieck polynomials, and give a new formula for the product of a dominant line bundle and a Schubert class in K-theory; a previous formula of this type was given by W. Fulton and A. Lascoux, and later generalized by H. Pittie and A. Ram. Part of this work is joint with F. Sottile.


Speaker's Contact Info: lenart(at-sign)csc.albany.edu


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