The Knuth-Robinson-Schensted correspondence and the Weak Polynomial Identities of M₁,₁(E)

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In this paper it is proved that the ideal Iw of the weak polynomial identities of the superalgebra M₁,₁(E) is generated by the proper polynomials [x₁,x₂,x₃] and [x₂,x₁][x₃,x₁][x₄,x₁]. 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 ∩ Iw. 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.