\magnification=1200
\hsize=4.25in
\vsize=44in
\overfullrule=0pt
\input amssym
\nopagenumbers
\noindent
%
%
{\bf Marc A. A. van Leeuwen}
%
%
\medskip
\noindent
%
%
{\bf Spin-Preserving Knuth Correspondences for Ribbon Tableaux}
%
%
\vskip 5mm
\noindent
%
%
%
%
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~$\{0,1\}^r$.  \par 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.  \par 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.

\bye

.
