Knuth-Robinson-Schensted对应与$M_{1,1}(E)$的弱多项式恒等式
The Knuth-Robinson-Schensted correspondence and the Weak Polynomial Identities of $M_{1,1}(E)$
摘要：
本文证明了超代数$M_{1,1}(E)$的弱多项式恒等式的理想$I_w$是由真多项式$[x_1,x_2,x_3]$和$[x_2,x_1][x_3,x_1][x_4,x_1]$生成的.这是证明任何无限域$F$的特征不同于2。如果$B$是$F< X>$的真多项式的子代数，则确定了商代数$B / B \cap I_w$的多齐次分支的基和维数.我们也计算这个代数的希尔伯特级数。本文的主要工具之一是我们发现的Knuth Robinson Schensted对应定义为单半标准tableaux的双重形状的一个变种。
英文摘要：
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.
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