In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system.
Background If the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal torus T, then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with André Weil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by Bernstein, Gelfand & Gelfand (1973) and independently Demazure (1973) as a tool to understand the Schubert calculus of the flag manifold. The Kostant polynomials are related to the Schubert polynomials defined combinatorially by Lascoux & Schützenberger (1982) for the classical flag manifold, when G = SL(n,C). Their structure is governed by difference operators associated to the corresponding root system. Steinberg (1975) defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of the weight lattice. If K is simply connected, this ring can be identified with the representation ring R(T) and the W-invariant subring with R(K). Steinberg's basis was again motivated by a problem on the topology of homogeneous spaces; the basis arises in describing the T-equivariant K-theory of K/T.
Definition Let Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ+ be a set of positive roots and Δ the corresponding set of simple roots. If α is a root, then sα denotes the corresponding reflection operator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ gives rise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products of simple root reflection. The minimal length for an element s is denoted
ℓ ( s ) {\displaystyle \ell (s)} . Pick an element v in V such that α(v) > 0 for every positive root. If αi is a simple root with reflection operator si
s i x = x − 2 ( x , α i ) ( α i , α i ) α i , {\displaystyle s_{i}x=x-2{(x,\alpha _{i}) \over (\alpha _{i},\alpha _{i})}\alpha _{i},}
then the corresponding divided difference operator is defined by
δ i f = f − f ∘ s i α i . {\displaystyle \delta _{i}f={f-f\circ s_{i} \over \alpha _{i}}.}
If ℓ ( s ) = m {\displaystyle \ell (s)=m} and s has reduced expression
s = s i 1 ⋯ s i m , {\displaystyle s=s_{i_{1}}\cdots s_{i_{m}},}
then
δ s = δ i 1 ⋯ δ i m {\displaystyle \delta _{s}=\delta _{i_{1}}\cdots \delta _{i_{m}}}
is independent of the reduced expression. Moreover
δ s δ t = δ s t {\displaystyle \delta _{s}\delta _{t}=\delta _{st}}
if ℓ ( s t ) = ℓ ( s ) + ℓ ( t ) {\displaystyle \ell (st)=\ell (s)+\ell (t)} and 0 otherwise. If w0 is the longest element of W, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then
… excerpt ends here. Continue reading the full article.
