Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux

We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2-31.Comment: 37 pages, 4 figures, new references adde

An affine generalization of evacuation

We establish the existence of an involution on tabloids that is analogous to Schutzenberger's evacuation map on standard Young tableaux. We find that the number of its fixed points is given by evaluating a certain Green's polynomial at $q = -1$, and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure

The cyclic sieving phenomenon: a survey

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes suggested by colleagues and the referee. To appear in the London Mathematical Society Lecture Note Series. The third version has a few smaller change

Major Index over Descent Distributions of Standard Young Tableaux

This thesis concerns the generating functions $f_{\lambda, k}(q)$ for standard Young tableaux of shape $\lambda$ with precisely $k$ descents, aiming to find closed formulas for a general form given by Kirillov and Reshetikhin in 1988. Throughout, we approach various methods by which further closed forms could be found. In Chapter 2 we give closed formulas for tableaux of any shape and minimal number of descents, which arise as principal specializations of Schur functions. We provide formulas for tableaux with three parts and one more than minimal number of descents, and demonstrate that the technique is extendable to any number of parts. In Chapter 3 we aim to reduce the complexity of Kirillov and Reshetikhin\u27s formula by identifying the summands contributing a nonzero amount to the polynomial. While the resulting formulas are lengthy, they greatly reduce the computation time for specified partition shapes and numbers of descents. In Chapter 4 we investigate an apparent relation among $f_{\lambda, k}(q)$ and $f_{\lambda, k-1}(q)$ and discuss how this may lead to a greater insight of the distribution of these statistics. Included appendices give a library of utilities in SageMath and Mathematica to generate the polynomials $f_{\lambda, k}$ and demonstrate Chapter 4\u27s relationships

Combinatorial Hopf Algebras, Noncommutative Hall-Littlewood Functions, and Permutation Tableaux

We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2-31.Mathematic

Cyclic sieving, skew Macdonald polynomials and Schur positivity

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is a multiple of $n$, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under a cyclic shift of the columns. The corresponding CSP polynomial is given by $E_{\lambda}(x;q;0)$. In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B.~Rhoades. We also introduce a skew version of $E_{\lambda}(x;q;0)$. We show that these are symmetric and Schur-positive via a variant of the Robinson--Schenstedt--Knuth correspondence and we also describe crystal raising- and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials

Quantum and affine Schubert calculus and Macdonald polynomials

This thesis is on the theory of symmetric functions and quantum and affine Schubert calculus. Namely, it establishes that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. Through this discovery, there is a construction of two one-parameter families of functions that respectively transition positively with Hall-Littlewood polynomials and Macdonald's P-functions. Furthermore, these functions specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. This shows that the theory of symmetric Macdonald polynomials connects with affine Schubert calculus.There is a generalization of the discovery of the strong order chains. This generalization connects the theory of Macdonald polynomials to quantum Schubert calculus. In particular, the approach leads to conjecture that all elements in a defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can be formulated as strong covers.Ph.D., Mathematics -- Drexel University, 201

Identities from representation theory

We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure