L. Lapointe, A. Lascoux, J. Morse
We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, S_λ[X(t-1)/(q-1)], can be reduced to addition in λ-rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, S_λ[X], and the monomial basis, m_λ[X].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider semisimple triangular operators acting in the symmetric component of the group algebra over the weight lattice of a root system. We present a determinantal formula for the eigenbasis of such triangular operators. This determinantal formula gives rise to an explicit construction of the Macdonald polynomials and of the Heckman-Opdam generalized Jacobi polynomials.
Correlation Functions of the Schur Process Through Macdonald Difference Operators
本文利用Schur多项式是Macdonald差分算子本征函数的性质，以新方法重新推导了Schur过程作为确定性点过程其关联函数为关联核矩阵子式的结论。
Macdonald operators are well known as the 'commutative family' acting on the symmetric functions over Q(q,t). If we suppose that q=exp(h) and t=exp(beta h) and observe the Taylor expansion around h=0, we can see the second-degree Dunkl operator appear especially as the coefficient of h^2. These Dunkl operators also consist of commutative family. Then, as to the coefficient of h^3, it is natural to expect that third-degree Dunkl operator appears. The object of this paper is to calculate the coefficients of h^3 in the h-expansion of Macdonald operators explicitly, to introduce the method of calc…
A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle and Schlosser's formula for partitions with length three.
The determinantal form of the partition function of the 6-vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur polynomial. Caradoc, Foda and Kitanine computed the partition function of the higher spin generalization of the 6-vertex model. In the present work it is shown that for a special value of the crossing parameter, referred to as the combinatorial point, the partition function reduces to a Macdonald polynomial.
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free fiel…
Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel develo…
In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially.
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.
A general method of deducing the determinantal expressions for polynomial and its derivative
该研究提出推导多项式及其导数行列式表达式的通用方法，以欧拉多项式为例给出相关行列式表达式，利用Gessel的函数方程建立二阶欧拉多项式及其导数的行列式表达式。
In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type extensions. Among these the $BC$-type interpolation Jack polynomials were probably unobserved until now. Much emphasis is put on combinatorial formulas and binomial formulas for (most of) these polynomials. Possibly new results derived from these formulas are a limit from Koornwinder to Macdonald polynomials, an explicit formula for Koornwinder polynomials in two…
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and $q$-Whittaker polynomials.
Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.
For permutations $v,w \in \mathfrak S_n$, Macdonald defines the skew divided difference operators $\partial_{w/v}$ as the unique linear operators satisfying $\partial_w(PQ) = \sum_v v(\partial_{w/v}P) \cdot \partial_vQ$ for all polynomials $P$ and $Q$. We prove that $\partial_{w/v}$ has a positive expression in terms of divided difference operators $\partial_{ij}$ for $i<j$. In fact, we prove that the analogous result holds in the Fomin-Kirillov algebra $\mathcal E_n$, which settles a conjecture of Kirillov.
Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures
本文利用Faà di Bruno公式、偏Bell多项式恒等式及比值求导公式，系统建立了Clark--Ismail两个猜想中相关函数的闭式公式、行列式表达式、递推关系、幂级数与特殊值，并进一步得到Hessenberg矩阵行列式公式及相关数列的对数凸性。
Rank recursion for $q$-Whittaker and Macdonald operators
本文引入并证明 $q$-Whittaker 和 Macdonald 算子的秩递推关系，用 $q$-变形二项分布表示 $q$-Whittaker 算子的 $k$ 次幂，并用 Cauchy 行列式表达 Macdonald 算子的秩递推。
Determinantal representations of alternating run polynomials
本文基于函数商高阶导数的行列式公式，推导了Eulerian多项式、Andre多项式及多种交错游程多项式的行列式表示，并揭示了其递推关系及与对偶Stirling排列的联系。
We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the…
Generalized Macdonald functions and quantum toroidal gl(1) algebra
本研究将普通Macdonald对称函数的$e_1$-Pieri规则、与Whittaker向量相关恒等式等公式推广到量子环面$\rm \frak{gl}(1)$代数对应的广义情形，提出五项关系与Fourier/Hopf配对的推广，并证明了广义Macdonald核的因子化猜想。
Jacobi-Trudi formulas and determinantal varieties
本文借助 Zelevinsky 的 BGG 复形构造阐明 Gessel 的 Schur 函数行列式表达式与经典 Jacobi-Trudi 公式的共同机制，并将该公式向多个方向推广。
Towards combinatorial derivations of K-polynomials for determinantal varieties
本文针对行列式簇的K多项式组合推导问题，提出对秩不超过k的复矩阵簇希尔伯特级数相关公式给出直接组合证明的目标，在k=1或k=min{m,n}-1时通过Littlewood–Richardson表征集上的显式符号反转对合完成了证明。
We examine the non-symmetric Macdonald polynomials $E_λ(x;q,t)$ at $q=1$, as well as the more general permuted-basement Macdonald polynomials. When $q=1$, we show that $E_λ(x;1,t)$ is symmetric and independent of $t$ whenever $λ$ is a partition. Furthermore, we show that for general $λ$, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of $t$, while the non-symmetric part only depends on the relative order of the entries in $λ$. We also examine the case $q=0$, which give rise to so called permuted-basement $t$-atoms. We prove expansion-…
Monomial expansions of modified Macdonald polynomials
本文发现十六个 Young 图填充统计量，给出修正 Macdonald 多项式的新组合公式，并由此建立四个紧凑公式和四个单项式展开显式表达式，其中一个与 Garbali-Wheeler 公式一致。
We compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at up to two marked points. When the Brill-Noether number $ρ$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $ρ=1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-Martín-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degen…
The asymmetric five vertex model on a rectangle
本文推导了矩形几何中具有任意底部和顶部边界条件的非齐次非对称五顶点模型的行列式表达式，解决了标准不相交格点路径方法不适用的问题。
We use the residue theorem to derive an expression for the number of lattice oints in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be u…
We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.
This talk is a report on joint work with A. Vaintrob [arXiv:math.CO/0109104 and math.GT/0111102]. It is organised as follows. We begin by recalling how the classical Matrix-Tree Theorem relates two different expressions for the lowest degree coefficient of the Alexander-Conway polynomial of a link. We then state our formula for the lowest degree coefficient of an algebraically split link in terms of Milnor's triple linking numbers. We explain how this formula can be deduced from a determinantal expression due to Traldi and Levine by means of our Pfaffian Matrix-Tree Theorem [arXiv:math.CO/0109…
We give an elementary symmetric function expansion for $MΔ_{m_γe_1}Πe_λ^{\ast}$ and $MΔ_{m_γe_1}Πs_λ^{\ast}$ when $t=1$ in terms of what we call $γ$-parking functions and lattice $γ$-parking functions. Here, $Δ_F$ and $Π$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M Δ_{Fe_1} Θ_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis $\{Πe_λ^\ast\…
Elliptic Deformation of the Gaiotto-Rapčák Corner VOA and the Associated Partially Symmetric Polynomials
本文构造椭圆 Miura 变换以表达椭圆角落 VOA 的流，证明用于其二次关系的组合公式，并猜想关联函数对应部分对称多项式，椭圆 Macdonald 多项式为其特例。
A new antisymmetrizer-to-determinant formula and a conjecture of Colomo and Pronko
本文建立了新的反对称化子-行列式恒等式，证明了Lukas Riegler等人的猜想，阐述了相关反对称化子，为解决Colomo与Pronko猜想提供了适配版本。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。