$W$-graph versions of tensoring with the $\S_n$ defining representation

We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = \S_n$. Our main application is to give two $W$-graph versions of tensoring with the $\S_n$ defining representation $V$, one being \H \tsr_{\H_J} - for \H, \H_J the Hecke algebras of $\S_n, \S_{n-1}$ and the other (\pH \tsr_{\H} -)_1, where $\pH$ 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 $V \tsr V \tsr - \to S^2 V \tsr -$. This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter $\u \to 0$. We make this approximation combinatorially explicit by determining it on cells. Also of interest is a combinatorial conjecture stating the restriction of \H to \H_J is "weakly multiplicity-free" for $|J| = n-1$, and a partial determination of the map \H \tsr_{\H_J} \H \xrightarrow{\counit} \H on canonical basis elements, where $\counit$ is the counit of adjunction.