T. Kyle Petersen, Pavlo Pylyavskyy, Brendon Rhoades
We show that Schützenberger's promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.
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Promotion and cyclic sieving on families of SSYT
该研究针对几类半标准杨表，验证其在提升操作下的循环筛分现象，提出计算斜半标准杨表电荷的新方法，推广了此前相关研究结果。
We prove a cyclic sieving result for the set of $3 \times k$ packed increasing tableaux with maximum entry $m :=3+k$ under K-promotion. The "curiosity" is that the sieving polynomial arises from the $q$-hook formula for standard tableaux of "toothbrush shape" $(2^3, 1^{k-2})$ with $m+1$ boxes, whereas K-promotion here only has order $m$.
Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks
研究高度为二的阶梯平面划分提升作用的循环筛选结果，利用自旋晶体张量幂解释及电网络空间坐标环二次部分的灌木丛基证明，还探讨了所有度数上电典范基存在时对所有高度阶梯平面划分提升的循环筛选意义。
We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on r…
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMah…
Let $δ=(δ_1,\ldots,δ_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $δ$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if $δ_i=h$ and a vertical strip if $δ_i=v$. We define $δ$-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that $δ$-promotion has period $n$ on rectangular $δ$-semistandard tableaux, generalizing the results of Haiman and Rhoades for standard and semistandard tableaux. The second main result states that the set of rectangular $δ$-semistandard tableau…
We give a new cyclic sieving phenomenon for semistandard Young tableaux $SSYT(λ,μ)$ of shape $λ=(m,n^b)$ and content $μ$, a $(b+2)$-tuple. We prove that $(SSYT(λ,μ),\langle \partial^{b+2} \rangle, f(q))$ exhibits the cyclic sieving phenomenon, where $\partial$ is the jeu de taquin promotion operator and $f(q)$ is a modified Kostka-Foulkes polynomial $\widetilde{K}_{λ,μ}(q)$, up to a power of $q$.
We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization, the shape of stabilized tableaux, and tableau stabilization as a permutation statistic. We apply tableau stabilization to construct the sufficiently large rectangular tableaux fixed by powers of promotion, which were counted by Brendon Rhoades via the cyclic sieving phenomenon
Skew characters and cyclic sieving
该研究将矩形标准杨表的循环分块现象结果推广至满足伪次数多项式单位根取值非负的斜标准杨表，通过边界条杨表数界的方法确定斜特征承载循环群置换表示的情形，并应用于一般线性群伴随表示张量幂不变量理论，证明置换与交错杨表间存在与旋转、晋升交错的双射。
Super Major Index Cyclic Sieving
本文研究了矩形形状的带符号标准 Young 表的循环筛分现象，结合 Schützenberger 提升和符号循环移位，证明其由超主指数生成函数描述，扩展了 Rhoades 的经典结果，并推广到任意非矩形形状。
In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(λ)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $λ$ is a Young diagram with $\ell(λ) < n$ and $\gcd( n, |λ| )=1$, then the triple $\left( \mathsf{SST}_n(λ), \mathsf{C}, q^{- κ(λ)} s_λ(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and sho…
Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins
本文研究路径图上置换环面提升算子，通过引入破碎提升算子并以液团滑动、石子滑动和硬币碰撞等模型刻画其动力学，完全确定了所有这类算子的阶并证明其轨道结构满足循环筛分现象。
We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degre…
A combinatorial expression for the coefficient of the Schur function $s_λ$ in the expansion of the plethysm $p_{n/d}^d \circ s_μ$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $λ$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_μ, s_λ \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_μ^n, s_λ \rangle$-element set under the $d^{\text{th}}$ power of an order-$n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $λ$ is rectangular, …
Cyclic sieving on permutations -- an analysis of maps and statistics in the FindStat database
本研究借助SageMath检索FindStat数据库，系统分析排列统计量与双射映射的循环筛选现象，新证明34个CSP实例、提出3个猜想，还发现同态性作用与CSP候选并无必然对应关系。
Promotion digraphs
研究为任意形状的标准和递增表格引入提升有向图，证明任意形状标准表格可由其唯一确定，猜测矩形递增表格也如此并给出证据，还用三行矩形递增表格的提升有向图猜测其与火烈鸟网的联系。
Orbit lengths for promotion on 2-row and near-hook tableaux
研究了2行和近钩表板上推广操作的轨道长度，发现其与n的多项式相关，并通过循环筛现象描述。
Rotation invariant webs for three row flamingo Specht modules
本文为 Specht 模块 $S^{(d^3, 1^{n-3d})}$ 引入了由正常 plabic 图索引的旋转不变 web 基，通过 skein 关系组合解释了对称群作用，并由此导出了钩长公式 $q$-模拟的循环筛分结果。
Kreweras words are words consisting of n A's, n B's, and n C's in which every prefix has at least as many A's as B's and at least as many A's as C's. Equivalently, a Kreweras word is a linear extension of the poset ${\sf V}\times [n]$. Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger's promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word …
Promotion permutations for tableaux
本文提出了轮换排列和轮换矩阵的新概念，并在涵盖多种已知表字类的更一般的波动表字背景下，全面阐述了其基础组合学与表示理论，以支持配套论文中关于 $SL_4$-web 基的归纳论证。
Evacuation of rectangular standard Young tableaux corresponds to reflection of $\mathfrak{sl}_n$ webs
本文证明矩形标准杨表的驱逐等价于相应 $\mathfrak{sl}_n$ 编织图在边翻转关系下的反射，推广了已有结果，并借助多色非交叉匹配建立联系。
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