We construct the Schubert basis of the torus-equivariant K-homology of the ane Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G = SLn, the K-homology of the ane Grassmannian is identied with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called K-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the ane stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick ane ag manifold.
Morse @cite_10 gives a combinatorial definition of the affine stable Grothendieck polynomials @math in terms of affine set-valued tableaux and also proves the Pieri rule for @math . The original @math -Schur functions @math @cite_19 @cite_14 , which arose in the study of Macdonald polynomials, involve a parameter @math . It appears that a @math -analogue @math of @math exists , defined in a similar manner to [Conjecture 9.11] LLMS . The connection of @math and Macdonald theory is explored in @cite_36 .
@cite_36 @cite_19 @cite_14 @cite_10
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Abstract We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k -Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k - K -Schur functions – Schubert representatives for the K -theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules. We prove that the Lam-Shimozono ''down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of non-commutative k-Schur functions of ''near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood-Richardson coefficients. The k-Young lattice Yk is a partial order on partitons with no part larger than k. This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103-146) from the study of the k-Schur functions sλ(k), Symmetric functions that from a natural basis of the space spanned by homogeneous funtions indexed by k-bounded partitions. The chains in the k-Young lattice are induced by a Pieritype rule experimentally satisfied by the k-Schur functions. Here, using a natural bijection between k-bounded partitons and k + 1-cores, we establish an algorithm for identifying chains in the k- Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric gruop S˜k+1 by a maximal parabolic subgruop. From this, the conjectured k-Pieri rule implies that the k-Kostka matrix connecting the homogeneous basis hλ λ∈Yk to sλ(k) λ∈Yk may now be obtained by counting appropriate classes of tableaux on k + 1-cores. This suggests that the conjecturally positive k-Schur expansion coefficients for Macdonald polynomials (reducing to q, t-Kostka polynomials for large k) could be described by a q, t-statistic on these tableaux, or equivalently on reduced words for affine permutations. The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain themes appearing in both theories that closely resemble each other. In certain cases, the two threads can be joined, leading to new insights beyond the classical realm of one particular theory.Discrete Morse theory provides combinatorial equivalents of several core concepts of classical Morse theory, such as discrete Morse functions, discrete gradient vector fields, critical points, and a cancelation theorem for the elimination of critical points of a vector field. Because of its simplicity, it not only maintains the intuition of the classical theory but allows to surpass it in a certain sense by providing explicit and canonical constructions that would become quite complicated in the smooth setting.Persistent homology quantifies topological features of a function. It defines the birth and death of homology classes at critical points, identifies pairs of these (persistence pairs), and provides a quantitative notion of their stability (persistence).Whereas (discrete) Morse theory makes statements about the homotopy type of the sublevel sets of a function, persistence is concerned with their homology. While homology is an invariant of homotopy equivalences, the converse is not true: not every map inducing an isomorphism in homology is a homotopy equivalence. In this thesis we establish a connection between both theories and use this combination to solve problems that are not easily accessibly by any single theory alone. In particular, we solve the problem of minimizing the number of critical points of a function on a surface within a certain tolerance from a given input function.