Schubert polynomials and skew Schur functions (Q1199818)

The author recalls the definitions of Schubert polynomials and successively introduces the skew Schur polynomials. They are defined as follows: let \(I=(i_ 1,\dots,i_ p)\) and \(J=(j_ 1,\dots,j_ p)\) be two \(p\)-tuples of integers such that \(0\leq i_ 1\leq\cdots\leq i_ p\), \(0\leq j_ 1\leq\cdots\leq j_ p\) and \(i_ k\leq j_ k\) for every \(k=1,\dots,p\). Then the skew Schur polynomial in the alphabet \(A_ n=\{a_ 1,\dots,a_ n\}\), \(S_{I/J}(A_ n)\), is the determinant of the matrix \((S_{j_ k+k-(j_ h+h)}(A_ n)_{h,k})\), where \(S_ m(A_ n)\) denotes the completely symmetric polynomial of degree \(m\) in the alphabet \(A_ n\). The author shows that any skew Schur polynomial is a restriction of a Schur polynomial. Finally, he shows an algorithm which computes a decomposition of a skew Schur function into a sum of Schur functions.