Quasi-polynomials of Capelli.~II

S. Yu. Antonov ; A. V. Antonova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 4-16

Résumé

This paper observes the continuation of the study of a certain kind of polynomials of type Capelli (Capelli quasi-polynomials) belonging to the free associative algebra $F\{X\bigcup Y\}$ considered over an arbitrary field $F$ and generated by two disjoint countable sets $X$ and $Y$. It is proved that if $char F=0$ then among the Capelli quasi-polynomials of degree $4k-1$ there are those that are neither consequences of the standard polynomial $S^-_{2k}$ nor identities of the matrix algebra $M_k(F)$. It is shown that if $char F=0$ then only two of the six Capelli quasi-polynomials of degree $4k-1$ are identities of the odd component of the $Z_2$-graded matrix algebra $M_{k+k}(F)$. It is also proved that all Capelli quasi-polynomials of degree $4k+1$ are identities of certain subspaces of the odd component of the $Z_2$-graded matrix algebra $M_{m+k}(F)$ for $m>k$. The conditions under which Capelli quasi-polynomials of degree $4k+1$ being identities of the subspace $M_1^{(m,k)}(F)$ are given.