Cyclic Sieving and Plethysm Coefficients

International audience A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ 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 $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Une expression combinatoire pour le coefficient de la fonction de Schur $s_{\lambda}$ dans l’expansion du pléthysme $p_{n/d}^d \circ s_{\mu}$ est donné pour tous $d$ que disent $n$, dans les cas où $n=2$, ou $\lambda$ est rectangulaire. Dans ces cas, le coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ se montre à compter, où l’on ignore le signe, le nombre des point fixés d’un ensemble de $\langle s_{\mu}^n, s_{\lambda} \rangle$ éléments sous la puissance $d^e$ d’une action cyclique de l’ordre $n$. Si $n=2$, l’action est l’involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si $\lambda$ est rectangulaire, l’action est une certaine puissance de l’avancement jeu-de-taquin de Schützenberger et Shimozono.Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas $n=2$ , la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d’une fonction de Schur.

An algorithm for generating ribbon tableaux and spin polynomials

arXiv:math.CO/0611824 International audience We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes.

Spin-Preserving Knuth Correspondences for Ribbon Tableaux

The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r\in{\bf N}_{>0}$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between $r$-coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). Viewing $r$-coloured permutations as matrices with entries in ${\bf N}^r$ (the non-zero entries being unit vectors), this correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core; the generalisation is derived from the RSK correspondence, again using the $r$-quotient map. Shimozono and White recently defined a more interesting generalisation of the Robinson-Schensted correspondence to $r$-coloured permutations and standard $r$-ribbon tableaux; unlike the Stanton-White correspondence, it respects the spin statistic on standard $r$-ribbon tableaux, relating it directly to the colours of the $r$-coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core, which respects the spin statistic on those tableaux in a similar manner, relating it directly to the matrix entries. We also define a similar generalisation of the asymmetric RSK correspondence, in which case the matrix entries are taken from $\{0,1\}^r$. More surprising than the existence of such a correspondence is the fact that these Knuth correspondences are not derived from Schensted correspondences by means of standardisation. That method does not work for general $r$-ribbon tableaux, since for $r\geq3$, no $r$-ribbon Schensted insertion can preserve standardisations of horizontal strips. Instead, we use the analysis of Knuth correspondences by Fomin to focus on the correspondence at the level of a single matrix entry and one pair of ribbon strips, which we call a shape datum. We define such a shape datum by a non-trivial generalisation of the idea underlying the Shimozono-White correspondence, which takes the form of an algorithm traversing the edge sequences of the shapes involved. As a result of the particular way in which this traversal has to be set up, our construction directly generalises neither the Shimozono-White correspondence nor the RSK correspondence: it specialises to the transpose of the former, and to the variation of the latter called the Burge correspondence. In terms of generating series, our shape datum proves a commutation relation between operators that add and remove horizontal $r$-ribbon strips; it is equivalent to a commutation relation for certain operators acting on a $q$-deformed Fock space, obtained by Kashiwara, Miwa and Stern. It implies the identity $$\sum_{\lambda\geq_r(0)}G^{(r)}_\lambda(q^{1\over2},X) G^{(r)}_\lambda(q^{1\over2},Y) =\prod_{i,j\in{\bf N}}\prod_{k=0}^{r-1}{1\over1-q^kX_iY_j}; $$ where $G^{(r)}_\lambda(q^{1\over2},X)\in{\bf Z}[q^{1\over2}][[X]]$ is the generating series by $q^{{\rm spin}(P)}X^{{\rm wt}(P)}$ of semistandard $r$-ribbon tableaux $P$ of shape $\lambda$; the identity is a $q$-analogue of an $r$-fold Cauchy identity, since the series factors into a product of $r$ Schur functions at $q^{1\over2}=1$. Our asymmetric correspondence similarly proves $$\sum_{\lambda\geq_r(0)}G^{(r)}_\lambda(q^{1\over2},X) \check G^{(r)}_\lambda(q^{1\over2},Y) =\prod_{i,j\in{\bf N}}\prod_{k=0}^{r-1}(1+q^kX_iY_j). $$ with $\check G^{(r)}_\lambda(q^{1\over2},X)$ the generating series by $q^{{\rm spin}^{\rm t}(P)}X^{{\rm wt}(P)}$ of transpose semistandard $r$-ribbon tableaux $P$, where ${\rm spin}^{\rm t}(P)$ denotes the spin as defined using the standardisation appropriate for such tableaux.