Mikhail Kogan, Abhinav Kumar
We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.
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Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's f…
Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above.
We give a direct proof of Schensted's lemma asserting that row and column insertion in a tableau commute.
We prove a positivity theorem for a certain family of operators defined in terms of the stochastic six-vertex model. We explore connections of this result with other vertex models and $t$-deformed Schensted insertions.
The purpose of this note is to give an insertion scheme proof of the formula, $$p_μ= \sum_{λ\vdash k} χ^λ(μ)s_λ,\formula$$ where $p_μ$ is the power sum symmetric function, $s_λ$ is the Schur function and $χ^λ(μ)$ is the irreducible character of the symmetric group $S_k$ indexed by the partition $λ$ and evaluated at a permutation of cycle type $μ=(μ_1,\ldots,μ_\ell)$. The proof of this formula is by direct application of the Robinson-Schensted-Knuth insertion scheme and a recent formula of Roichman.
We study some properties of domino insertion, focusing on aspects related to Fomin's growth diagrams. We give a self-contained proof of the semistandard domino-Schensted correspondence given by Shimozono and White, bypassing the connections with mixed insertion entirely. The correspondence is extended to the case of a nonempty 2-core and we give two dual domino-Schensted correspondences. We use our results to settle Stanley's `2^{n/2}' conjecture on sign-imbalance and to generalise the domino generating series of Kirillov, Lascoux, Leclerc and Thibon.
We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson-Schensted-Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.
In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently b…
We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian. These results are obtained by interpreting the Schubert bases of Gr combin…
Berele row-insertion and quantum symmetric pairs
本文通过AII型量子对称对为Berele行插入提供表示论解释，将其提升为余理想子代数上表示的同构，并建立AII型对偶RSK对应。
Cumulants of threshold for Schensted row insertion into random tableaux
本研究分析固定形状随机杨表的Schensted行插入阈值随机变量，给出其用Kerov转移测度表示、基于非交叉交替树求和的累积量组合公式，并证明大形状随机杨表第一行最右元依分布收敛于显式高斯分布。
We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-Schensted-Knuth Algorithm for semi-skyline augmented fillings. This procedure commutes with the RSK algorithm, and therefore retains many of its properties.
Fluctuations of Schensted row insertion
该研究针对随机输入下的Schensted行插入算法，分析插入新增方格位置的涨落，证明其渐近高斯性并给出均值方差表达式，同时提出RSK算法应用于长随机序列的相关猜想。
Fomin (1994) introduced a notion of duality between two graded graphs on the same set of vertices. He also introduced a generalization to dual graded graphs of the classical Robinson-Schensted-Knuth algorithm. We show how Fomin's approach applies to the binary search tree insertion algorithm also known as sylvester insertion, and to the hypoplactic insertion algorithm.
The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman-Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.
The numbers $f_λ$ of standard tableaux of shape $λ\vdash n$ satisfy 2 fundamental recursions: $f_λ= \sum f_{λ^-}$ and $(n + 1)f_λ=\sum f_{λ^+}$, where $λ^-$ and $λ^+$ run over all shapes obtained from $λ$ by adding or removing a square respectively. The first of these recursions is trivial; the second can be proven algebraically from the first. These recursions together imply algebraically the dimension formula $n! =\sum f_λ^2$ for the irreducible representations of $S_n$. We show that a combinatorial analysis of this classical algebraic argument produces an infinite family of algorithms, amon…
Eulerian insertion operators and an Eulerian form of the Pieri rule
该研究通过对称函数理论统一解释普通与主指标插入算子，推导欧拉插入算子表达式，建立对称函数环到插入算子代数的同态，得到皮耶里规则的欧拉形式。
The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order $\mathcal P$. In this paper, we define relations à la Knuth on the symmetric group for each $\mathcal P$ and conjecture that the associated $\mathcal P$-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, and Guay-Paquet. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of $\mathcal P$-tableaux that occur in the equivalence class. We prove …
The aim of this paper is to give a corrected bijective proof of Vershik's relations for the Kostka numbers. Our proof uses insertion and reverse insertion algorithms, as in the combinatorial proof of the Pieri rule.
We use Kang-Misra's combinatorial description of the crystal graphs for $U_{q}(G_{2})$ to introduce the plactic monoid for type $G_{2}$. Then we describe the corresponding insertion algorithm which yields a Schensted type correspondence. Next we give a simple algorithm for computing the canonical basis of any finite dimensional $U_{q}(G_{2})$-module.
We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur polynomials. To formulate these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show {certain} extended Jacobi-Trudi formulas for the Schur multiple zeta functions.
We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to q-Whittaker functions (or Macdonald polynomials with t=0) and reduces to the usual Robinson-Schensted algorithm when q=0. The q-insertion algorithm is `randomised', or `quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We…
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certain pairs of combinatorial objects (called pile configurations) that are most naturally defined in terms of generalized permutation pattern and barred pattern avoidance. This Extended Patience Sorting Algorithm is very similar to the Robinson-Schensted-Knuth (or RSK) Correspondence, which is itself built from repeated application of the Schensted Inse…
Using a generalization of the Schensted insertion algorithm to rc-graphs, we provide a Littlewood-Richardson rule for multiplying certain Schubert polynomials by Schur polynomials.
We give the first two-dimensional pictorial presentation of Berele's correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S) correspondence \cite{Robinson, Schensted} for the symplectic group $Sp(2n, \Cpx )$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,\Cpx)$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n, \Cpx)$. Two-dimensional …
We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and co…
Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial …
On the combinatorics of tableaux -- Graphical representation of insertion algorithms
本文针对多种表插入算法描述不一、关系隐晦的问题，利用图形化“插入图”刻画其特征，系统整理文献中的相关算法并形成可视化目录。
Insertion algorithms for Gelfand $S_n$-graphs
本文改进 Beissinger 的受限 RS 插入，提出在列末添加单元格的新双射，并利用两种插入算法分类一般对称群完美模型对应 $W$-图中的分子及猜想的胞腔。
We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset…
An insertion algorithm for multiplying Demazure characters by Schur polynomials
本文在Kohnert组合模型上引入一种推广RSK插入的算法，给出Demazure特征标与Schur多项式乘积的显式非负公式，部分解决Polo关于Demazure模张量积具有Schubert滤过的猜想。
Mixed jeu de taquin and a problem of Soojin Cho
本文提出了一种新的斜移位浆果Schur函数定义，并发展了一种新的雅格多游戏理论来计算混合插入。
In this paper, we introduce the ghost-box-ball system, which is an extended version of the classical soliton cellular automaton. It is initially motivated as a mechanism for making precise a connection between the Schensted insertion (of the Robinson-Schensted-Knuth correspondence) and the dynamical process of the box-ball system. In addition to this motivation, we explore generalisations of classical notions of the box-ball system, including the solitonic phenomenon, the asymptotic sorting property, and the invariant shape construction. We analyse the ghost-box-ball system beyond its initial …
We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_λf^λm_k^λ$, where the sum is over partitions $λ$ of $n$, $f^λ$ is the number of standard tableaux of shape $λ$, and $m_k^λ$ is the number of "vacillating tableaux" of shape $λ$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_λ(m_k^λ)^2$, where $B(2k)$ is the number of set partitions of $\{1, >..., 2k\}$. We show that this insertion restricts to work for th…
Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(Γ_s,Γ_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The graded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeled versions of the strong and weak orders of $W$ respectively. Using a construction of Lusztig for quivers with an admissible automorphism, we define folded insertion for a Kac-Moody algebra and obtain Sagan-Worley shifted insertion from Robinson-Schensted insertion as a special case. Drawing on work of Stembridge, we analyze the induced subgraphs of $(Γ_s,Γ_w)$ which are distri…
Despite having been introduced in 1962 by C.L. Mallows, the combinatorial algorithm Patience Sorting is only now beginning to receive significant attention due to such recent deep results as the Baik-Deift-Johansson Theorem that connect it to fields including Probabilistic Combinatorics and Random Matrix Theory. The aim of this work is to develop some of the more basic combinatorics of the Patience Sorting Algorithm. In particular, we exploit the similarities between Patience Sorting and the Schensted Insertion Algorithm in order to do things that include defining an analog of the Knuth relati…
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