The Knuth-Robinson-Schensted correspondence and the Weak Polynomial Identities of $M_{1,1}(E)$

In this paper it is proved that the ideal $I_w$ of the weak polynomial identities of the superalgebra $M_{1,1}(E)$ is generated by the proper polynomials $[x_1,x_2,x_3]$ and $[x_2,x_1][x_3,x_1][x_4,x_1]$. This is proved for any infinite field $F$ of characteristic different from 2. Precisely, if $B$ is the subalgebra of the proper polynomials of $F< X>$, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra $B / B \cap I_w$. We compute also the Hilbert series of this algebra. One of the main tools of the paper is a variant we found of the Knuth-Robinson-Schensted correspondence defined for single semistandard tableaux of double shape.