Newton polytopes of dual kk-Schur polynomials

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper we show that the support of each dual $k$-Schur polynomial indexed by a $k$-bounded partition coincides with that of the Schur polynomial indexed by the same partition, and hence the two polynomials share the same saturated Newton polytope. The main result is based on our recursive algorithm to generate a semistandard $k$-tableau for a given shape and $k$-weight. As consequences, we obtain the M-convexity of dual $k$-Schur polynomials, affine Stanley symmetric polynomials and cylindric skew Schur polynomials.Comment: 20 pages, 8 figure

Combinatorial Interpretations of Induced Sign Characters of the Hecke Algebra

Combinatorial interpretations have been used to show the total nonnegativity of induced trivial character and induced sign character immanants. The irreducible character immanants are known to be totally nonnegative as well, however, providing a combinatorial interpretation remains an open problem. To find such combinatorial interpretations we explore the quantum analogs of the symmetric group characters associated to the above mentioned immanants. In this paper, a combinatorial interpretation for the quantum induced sign characters on certain elements of the Hecke algebra is provided. This interpretation is then related to the quantum chromatic symmetric function introduced by Shareshian and Wachs. These interpretations involve a certain class of posets and associated planar networks. Lastly, for a restricted subset of these planar networks, properties of the sequence of coefficients of the induced sign characters of the Hecke algebra are discussed

Prism tableaux and alternating sign matrices

A. Lascoux and M.-P. Schutzenberger introduced Schubert polynomials to study the cohomology ring of the complete flag variety Fl(C^n). Each Schubert polynomial corresponds to the class defined by a Schubert variety X_w in Fl(C^n). Schubert polynomials are indexed by elements of the symmetric group and form a basis of the ring Z[x1,x2,...]. The expansion of the product of two Schubert polynomials in the Schubert basis has been of particular interest. The structure coefficients are known to be nonnegative integers. As of yet, there are only combinatorial formulas for these coefficients in special cases, such as the Littlewood-Richardson rule for multiplying Schur polynomials. Schur polynomials form a basis of the ring of symmetric polynomials. They have a combinatorial formula as a weighted sum over semistandard tableaux. In joint work with A. Yong, the author introduced prism tableaux. A prism tableau consists of a tuple of tableaux, positioned within an ambient grid. With A. Yong, the author gave a formula for Schubert polynomials using prism tableaux. We continue the study of prism tableaux, detailing their connection to the poset of alternating sign matrices (ASMs). Schubert polynomials can be interpreted as multidegrees of the matrix Schubert varieties of Fulton. We study a more general class of determinantal varieties, indexed by ASMs. More generally, one can consider subvarieties of the space of n by n matrices cut out by imposing rank conditions on maximal northwest submatrices. We show that, up to an affine factor, such a variety is isomorphic to an ASM variety. The multidegrees of ASM varieties can be expressed as a sum over prism tableaux. In joint work with A. Yong and R. Rimanyi, the author studies representations of quivers and their connection to the dilogarithm identities of M. Reineke. We give a bijective proof to establish an identity of generating series. This bijection uses a generalization of Durfee squares. From this identity, we give a new proof of M. Reineke's identities in type A

Webs for Flamingo Specht Modules

A web basis of a representation of $\mathfrak{S}_n$ is a basis of the representation for which the action of $\mathfrak{S}_n$ can be understood through combinatorial rules called skein relations. In this thesis, we study web bases for two families of irreducible $\mathfrak{S}_n$ representations, indexed by the partitions $(d^2, 1^{n-2d})$ and $(d^3, 1^{n-3d})$. The first was introduced by Rhoades and is indexed by noncrossing set partitions of $n$. We use it to give a model for the top degree component of the fermionic diagonal coinvariant ring, and introduce another similar basis to model the entire fermionic diagonal coinvariant ring. We also give an embedding of the noncrossing set partition representation into an induction product of the Temperley-Lieb representation with a sign representation, thereby providing alternate proofs that the skein relations which define the noncrossing set partition representation are in fact well defined. The second web basis is new, and simultaneously generalizes the $SL_3$ web basis of Kuperberg and the noncrossing set partition web basis. To define it and show it gives a basis, we draw on the combinatorics of Plabic graphs, jellyfish invariants, and weblike subgraphs

Stable-Limit Cherednik Theory

This thesis is centered around extending families of representation theoretic objects corresponding to finite rank GL to the setting of infinite rank GL. Specifically, we study representations of the double affine Hecke algebras in type GL, the elliptic Hall algebra, and the double Dyck path algebra. Throughout this thesis we will develop new methods for constructing representation theoretic objects from families of finite rank classical objects and ways to understand these representations.In the first chapter, we give an overview of the background information regarding Macdonald theory and Cherednik theory and of recent results in the area including the Shuffle Theorem. This chapter contains a review of the necessary algebraic, combinatorial, and representation theoretic definitions which will be used throughout the thesis. In Chapter 2, we investigate limits of non-symmetric Macdonald polynomials and their place in the theory of almost symmetric functions. We will construct a basis of simultaneous eigenvectors for the limit Cherednik operators of Ion-Wu and investigate many of their properties. Further, we construct new operators on the space of almost symmetric functions generalizing the higher delta operators in Macdonald theory. Lastly, we explicitly compute q,t specializations of this basis to find a generalization of Schur functions to the almost symmetric functions with interesting combinatorial and representation theoretic properties.Chapter 3 revolves around a family of modules called the Murnaghan-type representations for the elliptic Hall algebra generated using a stable-limit procedure from the vector-valued polynomial DAHA representations of Dunkl-Luque. This family of modules is indexed by partitions and generalizes the standard polynomial representation of EHA. We will construct a special family of generalized symmetric Macdonald functions as simultaneous eigenvectors for a generalized Macdonald operator and investigate their properties. Lastly, in Chapter 4 we will construct new representations of the double Dyck path algebra built from compatible families of DAHA representations. We will use this general procedure to define Murnaghan-type representations using the EHA representations in Chapter 2

Semistandard tableaux associated with generalized labellings of posets

We describe a correspondence between a family of labelled partially ordered sets and semi-standard Young tableaux. Moreover, we define some operations among labelled posets which naturally correspond to operations among the associated semistandard Young tableaux