Robinson-Schensted-Knuth correspondence and 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 ∩ I_w ). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape.