Spin-Preserving Knuth Correspondences for Ribbon Tableaux

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