Tableau ring

Definition

The tableau ring $R$ is defined as the following (noncommutative) ring with unity:

• As a $\mathbb {Z}$-module it is freely generated by basis elements corresponding to all the semistandard tableaux
• The multiplication operation is defined by linearly extending the multiplication of tableaux, which is defined on basis elements

The tableau ring is a $\mathbb {Z}$-algebra. Given any commutative ring with unity, we can also consider a tableau ring over $S$ by simply tensoring the tableau ring with $S$, treating both as $\mathbb {Z}$-modules.

The multiplicative identity is the empty tableau.

Sometimes instead of considering the whole tableau ring, we consider the subring $R_{m}$ which is the subring generated by those tableaux with entries only uptil $m$.

Facts

The canonical map to the polynomial ring

There is a canonical map from the tableau ring to the polynomial ring over $\mathbb {Z}$, in countably many variables. Given, a tableau $T$, define the weight as the ordered partition $(\mu _{1},\mu _{2},\ldots )$ where $\mu _{i}$ is the number of occurrences of $i$ in the tableau. Then to the tableau, associate the polynomial $\prod _{i}x_{i}^{\mu _{i}}$.

This mapping extends to a ring homomorphism from the tableau ring to the polynomial ring. Note that this homomorphism forgets much of the tableau structure since it maps tableaux of the same content, to the same monomial.

Note that this homomorphism has the further property that it sends $R_{m}$ to the polynomial ring in $m$ variables.