A new class of symmetric polynomials defined in terms of tableaux
Let z signify a set of n indeterminates, and let \(\lambda\) be a partition having n parts \(\geq 0\). The authors discuss the connection between Young- Weyl standard tableaux of shape \(\lambda\) and Gel'fand-Weyl patterns (m) whose top row is \(\lambda\), and then define new symmetric polynomials \(t_{\lambda}(z)\) as a sum over (m) of products over i and j of rising factorials \((x)_ a=x(x+1)...(x+a-1)\) with \(x=(z_ j-m_{ij}-j+i+1)\) and \(a=m_{ij}-m_{ij-1}\). Remarkably, they are able to prove that such polynomials are indeed symmetric in the indeterminates \((z_ 1,z_ 2,...,z_ n)\) and that they form a \({\mathbb{Z}}\)-basis of the corresponding ring of symmetric polynomials. They are also shown to form a natural basis for the expansion of \(\prod^{n}_{i=1}(x+1-z_ i)_ k.\) Various other properties of \(t_{\lambda}(z)\) are established, and two generalizations are given, namely \(T_{\lambda}(\alpha;z)\) and \(t_{\lambda /\mu}(z)\). For \(\alpha =-x-y-k-1\) it is shown that the former constitute a natural basis for the expansion of \(\prod^{n}_{i=1}(x+1-z_ i)_ k(y+1-z_ i)_ k,\) and the latter are related to skew tableaux. Both are symmetric polynomials. In the case of \(T_{\lambda}(\alpha;z)\) a proof of this is provided, but in the case of \(t_{\lambda /\mu}(z)\) a proof is deferred to a subsequent paper. Some tabulations of \(t_{\lambda}(z)\) are given for \(n=1,2\) and 3. For those more familiar with Young-Weyl standard tableaux than with Gel'fand-Weyl patterns it may be of interest to note that \(t_{\lambda}(z)\) may, the reviewer believes, be recast in the form of a sum over standard tableaux of a product of factors, one for each entry. An entry k in the ith row and jth column being represented by the factor \((z_ k-k+i-j+1)\).
scientific article; zbMATH DE number 4143983
scientific article