Quasi-polynomials of Capelli. III

Abstract

In this paper polynomials of Capelli type (double and quasi-polynomials of Capelli) belonging to a free associative algebra $F\{X\cup Y\}$ considering over an arbitrary field $F$ and generated by two disjoint  countable  sets $X, Y$  are investigated.  It  is shown  that  double Capelli's  polynomials $C_{4k,\{1\}}$, $C_{4k,\{2\}}$ are consequences of the standard polynomial $S^-_{2k}$. Moreover, it  is  proved that  these  polynomials equal to zero both for square and for rectangular matrices of corresponding  sizes. In this paper it is also shown that all Capelli's quasi-polynomials of the $(4k+1)$ degree are minimal identities of odd component of $Z_2$-graded matrix algebra $M^{(m, k)}(F)$ for any  $F$ and $m\ne k$.

Keywords

• t-ideal
• standard polynomial
• capelli polynomial