The ring of symmetric functions $Λ$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on $Λ$ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood--Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. In the first part of this thesis we show that by using a generalization of the classical umbral calculus of Gian-Carlo Rota, one may deform the basis of Schur functions to find many other bases for which the Littlewood--Richardson numbers as coproduct structure constants. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These "quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These quasisymmetric Macdonald polynomials refine the (symmetric) Macdonald polynomials and specialize at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg. This is an extended abstract.
Set-valued tableaux formulas play an important role in Schubert calculus. Using the box greedy reduced word for the construction of the Macdonald polynomials, we convert the alcove walk formula for Macdonald polynomials to a set-valued tableaux formula for Macdonald polynomials. Our hope is that providing set-valued tableaux formulas for Macdonald polynomials will help to strengthen the analogies and possible connections between the calculus of Macdonald polynomials and Schubert calculus.
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.
We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-$t$-Macdonald polynomials of two rows.
Macdonald characters from a new formula for Macdonald polynomials
Université de Lorraine（洛林大学）；CNRS, IECL（法国国家科学研究中心，IECL）；Université de Paris（巴黎大学）；CNRS, IRIF（法国国家科学研究中心，IRIF）；Harvard University（哈佛大学）；Università di Pisa（比萨大学）
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials…
We demonstrate the validity of previously conjectured explicit expressions for the norm and the evaluation of the Macdonald polynomials in superspace. These expressions, which involve the arm-lengths and leg-lengths of the cells in certain Young diagrams, specialize to the well known formulas for the norm and the evaluation of the usual Macdonald polynomials.
The quasisymmetic Macdonald polynomials $G_γ(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_λ(X; q, t)$ with the property that $G_γ(X; 0, 0)$ equals $QS_γ(X)$, the quasisymmetric Schur polynomial of [9]. We derive an expansion for $G_γ(X; q, t)$ in the fundamental basis of quasisymmetric functions.
Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials
我们提出了代表阿斯基-威尔逊多项式的四重级数展开。为了得到这个结果，我们依次使用了几个基本超几何级数的求和和变换公式，包括Verma的域的q-扩张和Wimp展开式，Andrews的Watson的3F 2和的终止q-模拟，Singh的二次变换.作为应用，我们给出了BCn（n in Z>0）型Koornwinder多项式的一个列图显式公式.在参数特殊化的情况下，我们用一行图恢复了Bn，Cn和Dn型Macdonald多项式的Lassalle公式，从而证明了他的结论.
The Garsia--Haiman module is a bigraded $\mathfrak{S}_n$-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an $\mathfrak{S}_n$-set $X$ to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia--Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entrie…
This paper gives bijective proofs of some novel coinversion identities first discovered by Ayyer, Mandelshtam, and Martin (arxiv:2011.06117) as part of their proof of a new combinatorial formula for the modified Macdonald polynomials $\tilde{H}_μ$. Those authors used intricate algebraic manipulations of $q$-binomial coefficients to prove these identities, which imply the existence of certain bijections needed in their proof that their formula satisfies the axioms characterizing $\tilde{H}_μ$. They posed the open problem of constructing such bijections explicitly. We resolve that problem here.
This is a paper about $c$-functions and Macdonald polynomials. There are $c$-function formulas for $E$-expansions of $P_λ$ and $A_{λ+ρ}$, principal specializations of $P_λ$ and $E_μ$, for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from Macdonald (2003). In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials.
Recently James Martin introduced multiline queues, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a model of particles hopping on a one-dimensional lattice, which was introduced around 1970, and has been extensively studied in statistical mechanics, probability, and combinatorics. In this article we give an independent proof of Martin's result, and we show that by introducing additional statistics on multiline queues, we can use them to give a new combinatorial formu…
We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb{R}$ with non-negative values on Macdonald symmetric functions $P_λ$, that was conjectured by S.V. Kerov in 1992.
We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to $q,t$-analogs of the dimensions of the irreduc…
