Geodesic
$W$-graph versions of tensoring with the $\mathcal S_n$ defining representation.
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 545-585.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin (Math. Z. $244(2)$:415-431, 2003 and Manuscr. Math. $115(4)$:495-511, 2004), focusing on the case $W = S _{ n}$ W = mathcalS_n. Our main application is to give two $W$-graph versions of tensoring with the $S _{ n}$ mathcalS_n defining representation $V$, one being $S _{ n}, S _{ n -1}$ mathcalS_n, mathcalS_n-1 and the other $###$, where $###$ is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding $W$-graph versions of the projection $V\otimes V\otimes - \rightarrow S ^{2} V\otimes $- . This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter $u\rightarrow 0$. We make this approximation combinatorially explicit by determining it on cells and relate this to RSK growth diagrams.
Keywords: keywords W-graph, Hecke algebra, restriction and induction, canonical basis, growth diagram 
@article{JAC_2011__34_4_a8,
     author = {Blasiak, Jonah},
     title = {$W$-graph versions of tensoring with the $\mathcal S_n$ defining representation.},
     journal = {Journal of Algebraic Combinatorics},
     pages = {545--585},
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     volume = {34},
     number = {4},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a8/}
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Blasiak, Jonah. $W$-graph versions of tensoring with the $\mathcal S_n$ defining representation.. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 545-585. http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a8/