Michel Lassalle
We give an elementary proof of the development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions.
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Hadamard Products of dual Jacobi-Trudi matrices
研究Jacobi-Trudi矩阵阿达马乘积的正性性质，证明类带状斜形状下其Temperley-Lieb不变量为Schur正，肯定Sokal关于该类子式的猜想，提供表示论证明及Schur展开。
In 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and Liu introduced refined dual stable Grothendieck polynomials by putting additional sequence of parameters in the combinatorial formula of Lam and Pylyavskyy. Grinberg conjectured a Jacobi--Trudi type formula for refined dual stable Grothendieck polynomials. In this paper this conjecture is proved by using bijections of Lam and Pylyavskyy.
Total positivity of Hadamard product of dual Jacobi--Trudi matrices
本研究强化Wagner 1992年结果，建立Jacobi--Trudi矩阵Hadamard乘积的全单项式正性，解决Sokal猜想，给出带型指标该矩阵Hadamard平方的显式正Schur展开及表示论证明。
Let $\mathrm{JT}_λ$ be the Jacobi-Trudi matrix corresponding to the partition $λ$, so $\det\mathrm{JT}_λ$ is the Schur function $s_λ$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots=x_n=1$ and all other $x_i=0$. Then the entries of $\mathrm{JT}_λ$ become polynomials in $n$ of the form ${n+j-1\choose j}$. We determine the Smith normal form over the ring $\mathbb{Q}[n]$ of this specialization of $\mathrm{JT}_λ$. The proof carries over to the specialization $x_i=q^{i-1}$ for $1\leq i\leq n$ and $x_i=0$ for $i>n$, where we set $q^n=y$ and work over the ring $\mathbb{Q}(q)[y]$.
We use the Jacobi-Trudi formula to execute "explicit" evaluation of determinants of Stirling numbers of both kinds. We also offer a Maple package accompanying the paper on the personal websites at the end of the second page.
Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and…
Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$
本文通过BGG范畴O中的张量积重数重新证明斜Schur多项式的Jacobi-Trudi恒等式，并确立截断展开的Schur正性，同时解释系数为张量积重数。
In their study of characters of minimal affinizations of representations of orthogonal and symplectic Lie algebras, Chari and Greenstein conjectured that certain Jacobi-Trudi determinants satisfy an alternating sum formula. In this note, we prove their conjecture and slightly more. The proof relies on some symmetries of the ring of symmetric functions discovered by Koike and Terada. Using results of Hernandez, Mukhin-Young, and Naoi, this implies that the characters of minimal affinizations in types B, C, and D are given by a Jacobi-Trudi determinant.
In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_λ$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the…
Character expansions and affine Jacobi-Trudi identities
本文通过给出B_n、C_n和BC_n型多参数Hall--Littlewood多项式在矩形形状下的特征展开，证明了Ole Warnaar提出的六个仿射Jacobi-Trudi型恒等式。
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。
We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.
We give an interpretation of the boson-fermion correspondence as a direct consequence of Jacobi-Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi-Trudi identity the action of Clifford algebra on polynomial algebras that arrives as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions asso…
We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type C_n. Like the D_n case studied by the authors recently, applying the Gessel-Viennot path method with an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.
We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type D_n. Unlike the A_n and B_n cases, a simple application of the Gessel-Viennot path method does not yield a positive sum expression of the determinant over a set of tuples of paths. However, applying an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagra…
Multislant matrices and Jacobi--Trudi determinants over finite fields
本文研究有限域上Jacobi-Trudi矩阵行列式为零的概率，对部分构成等差数列的分拆给出公式并推广了Anzis等人的猜想，同时证明带状斜分拆情形下行列式在有限域上均匀分布。
Affine Jacobi-Trudi Identities and $q,t$-Rogers-Ramanujan Identities
猜想正交和辛Schur函数的对偶Jacobi-Trudi恒等式的仿射类比，用于推导Rogers-Ramanujan恒等式的t-模拟，包括经典、Andrews-Gordon及GOW恒等式，并证明任意高度矩形分区Schur函数的仿射对偶Jacobi-Trudi恒等式。
By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives an affirmative answer to an open problem posed by Kuperberg.
In the literature there are several determinant formulas for Schur functions: the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula, the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the Hamel-Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel-Goulden formula, w…
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…
Skew symplectic and orthogonal characters through lattice paths
针对斜辛与正交Schur函数行列式表达式研究较少的问题，研究者基于Koike和Terada表模型的格点路径描述，用组合方法建立其对偶Jacobi-Trudi与Lascoux-Pragacz公式的类似形式，并代数推导得普通Jacobi-Trudi公式。
We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant-type formula analogous to the classical definition of Schur polynomials.
We use Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity.
Quadratic relations for ninth variations of Schur functions and application to Schur multiple zeta functions
本文为Schur函数第九变体建立二次代数关系，并应用于研究对角常数Schur多重zeta函数在矩形形状下的特殊值。
We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial $\mathfrak S_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi-Trudi identity. For such $w$, this determinantal formula is equivalent to a (signed) subtraction-free expansion of $\mathfrak S_w$ in the basis of standard elementary monomials.
We derive a Jacobi-Trudi type formula for Jack functions of rectangular shapes. In this formula, we make use of a hyperdeterminant, which is Cayley's simple generalization of the determinant. In addition, after developing the general theory of hyperdeterminants, we give summation formulae for Schur functions involving hyperdeterminants, and evaluate Toeplitz-type hyperdeterminants by using Jack function theory.
The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let $Γ_{(t|X)}$ be the vertex operator defined by $Γ_{(t|X)} s_α=\sum_{n \in \mathbb{Z}} s_{(n,α)} [X] t^n$. We provide a combinatorial proof for the identity $Γ_{(t|X)} s_α= σ[tX] s_α\big[x-1/t\big] $ due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of $s_{(n,α)} [X] $ in the Schur basis, for any integer value of $n$.
Temperley-Lieb Crystals
本文针对 Temperley-Lieb 子式推广 Littlewood-Richardson 法则，引入洗牌 tableau，并利用 Stembridge 公理证明其构成 A 型 Kashiwara 晶体，从而给出相关 Schur 系数的组合解释。
Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity
本文提出了基于peelable tableaux和shuffle tableaux的Littlewood--Richardson系数新公式，可用于高效计算广义系数并恢复其对称性，同时证明了Schur对数凹性猜想的一个特例。
The Jacobi-Trudi formula implies some interesting quadratic identities for characters of representations of $gl_n$. Earlier work of Kirillov and Reshetikhin proposed a generalization of these identities to the other classical Lie algebras, and conjectured that the characters of certain finite-dimensional representations of $U_q(g-affine)$ satisfy it. Here we use a positivity argument to show that the generalized identities have only one solution.
We discover an explicit expansion formula for the powers $s$ of the Euler Product (or Dedekind $η$-function) in terms of hook lengths of partitions, where the exponent $s$ is any complex number. Several classical formulas have been derived for certain integers $s$ by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald. In particular, Macdonald obtained expansion formulas for the integer exponents $s$ for which there exists a semi-simple Lie algebra of dimension $s$. For the type $A_l^{(a)}$ he has expressed the $(t^2-1)$-st power of the Euler Product as a sum of weighted integer…
Temperley-Lieb Immanants, Key Positivity, and Demazure Crystals
本文将Schur正性结果推广到关键正性，借助Demazure晶体与洗牌表证明标记Jacobi-Trudi矩阵的Temperley-Lieb积和式关键正，给出标记斜Schur多项式乘积的关键展开规则、对数凹性结果及Demazure晶体新刻画。
The Jacobi-Trudi formulas imply that the minors of the banded Toeplitz matrices can be written as certain skew Schur polynomials. In 2012, Alexandersson expressed the corresponding skew partitions in terms of the indices of the struck-out rows and columns. In the present paper, we develop the same idea and obtain some new applications. First, we prove a slight generalization and modification of Alexandersson's formula. Then, we deduce corollaries about the cofactors and eigenvectors of banded Toeplitz matrices, and give new simple proofs to the corresponding formulas published by Trench in 198…
From total positivity to pure free resolutions
本研究以Jacobi-Trudi恒等式为基础建立全正整数序列理论与Koszul代数的平行关系，构造二次超曲面环的Schur模新类似物，并据此构建出与多项式环情形完全类似的纯自由分解。
Infinite-dimensional $q$-Jacobi Markov processes
本文构造了 $N$ 维 $q$-Jacobi 马尔可夫过程，并证明其在 $N\to\infty$ 时存在极限，得到无限维 Feller 过程，其证明依赖于大 $q$-Jacobi 多项式的对称函数提升。
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