Commun. Korean Math. Soc. 2000 Vol. 15, No. 1, 29-36

Woo Lee Kwangju University

Abstract : The $T$-ideal of $F \langle X \rangle$ generated by $x^n$ for all $x\in X$, is generated also by the symmetric polynomials. For each symmetric polynomial, %$S_n ( a_1, a_2, \dots, a_n )$, there corresponds one row of the incidence matrix. Finding the nilpotency of nil-algebra of nil-index $n$ is equivalent to determining the smallest integer $\mathcal{N}$ such that the $\langle n,\mathcal{N}\rangle$-incidence matrix has rank equal to $\mathcal{N} !$. In this work, we show that the $\langle n,\frac{n(n+1)}2\rangle^{(1, \dots , n)}$-incidence matrix is center-symmetric.

Keywords : incidence matrix, nil-algebra, center symmetry