In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by Lascoux & Schützenberger (1982) and are named after Hermann Schubert.
Background Lascoux (1995) described the history of Schubert polynomials. The Schubert polynomials S w {\displaystyle {\mathfrak {S}}_{w}} are polynomials in the variables x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } depending on an element w {\displaystyle w} of the infinite symmetric group S ∞ {\displaystyle S_{\infty }} of all permutations of N {\displaystyle \mathbb {N} } fixing all but a finite number of elements. They form a basis for the polynomial ring Z [ x 1 , x 2 , … ] {\displaystyle \mathbb {Z} [x_{1},x_{2},\ldots ]} in infinitely many variables. The cohomology of the flag manifold Fl ( m ) {\displaystyle {\text{Fl}}(m)} is Z [ x 1 , x 2 , … , x m ] / I , {\displaystyle \mathbb {Z} [x_{1},x_{2},\ldots ,x_{m}]/I,} where I {\displaystyle I} is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial S w {\displaystyle {\mathfrak {S}}_{w}} is the unique homogeneous polynomial of degree ℓ ( w ) {\displaystyle \ell (w)} representing the Schubert cycle of w {\displaystyle w} in the cohomology of the flag manifold Fl ( m ) {\displaystyle {\text{Fl}}(m)} for all sufficiently large m . {\displaystyle m.}
Properties If w 0 {\displaystyle w_{0}} is the permutation of longest length in S n {\displaystyle S_{n}} then S w 0 = x 1 n − 1 x 2 n − 2 ⋯ x n − 1 1 {\displaystyle {\mathfrak {S}}_{w_{0}}=x_{1}^{n-1}x_{2}^{n-2}\cdots x_{n-1}^{1}}
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