- Source: Kostant polynomial
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
δ
w
0
f
=
∑
s
∈
W
det
s
f
∘
s
∏
α
>
0
α
.
{\displaystyle \delta _{w_{0}}f={\sum _{s\in W}\det s\,f\circ s \over \prod _{\alpha >0}\alpha }.}
More generally
δ
s
f
=
det
s
f
∘
s
+
∑
t
<
s
a
s
,
t
f
∘
t
∏
α
>
0
,
s
−
1
α
<
0
α
{\displaystyle \delta _{s}f={\det s\,f\circ s+\sum _{t
for some constants as,t.
Set
d
=
|
W
|
−
1
∏
α
>
0
α
.
{\displaystyle d=|W|^{-1}\prod _{\alpha >0}\alpha .}
and
P
s
=
δ
s
−
1
w
0
d
.
{\displaystyle P_{s}=\delta _{s^{-1}w_{0}}d.}
Then Ps is a homogeneous polynomial of degree
ℓ
(
s
)
{\displaystyle \ell (s)}
.
These polynomials are the Kostant polynomials.
Properties
Theorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.
In fact the matrix
N
s
t
=
δ
s
(
P
t
)
{\displaystyle N_{st}=\delta _{s}(P_{t})}
is unitriangular for any total order such that s ≥ t implies
ℓ
(
s
)
≥
ℓ
(
t
)
{\displaystyle \ell (s)\geq \ell (t)}
.
Hence
det
N
=
1.
{\displaystyle \det N=1.}
Thus if
f
=
∑
s
a
s
P
s
{\displaystyle f=\sum _{s}a_{s}P_{s}}
with as invariant under W, then
δ
t
(
f
)
=
∑
s
δ
t
(
P
s
)
a
s
.
{\displaystyle \delta _{t}(f)=\sum _{s}\delta _{t}(P_{s})a_{s}.}
Thus
a
s
=
∑
t
M
s
,
t
δ
t
(
f
)
,
{\displaystyle a_{s}=\sum _{t}M_{s,t}\delta _{t}(f),}
where
M
=
N
−
1
{\displaystyle M=N^{-1}}
another unitriangular matrix with polynomial entries.
It can be checked directly that as is invariant under W.
In fact δi satisfies the derivation property
δ
i
(
f
g
)
=
δ
i
(
f
)
g
+
(
f
∘
s
i
)
δ
i
(
g
)
.
{\displaystyle \delta _{i}(fg)=\delta _{i}(f)g+(f\circ s_{i})\delta _{i}(g).}
Hence
δ
i
δ
s
(
f
)
=
∑
t
δ
i
(
δ
s
(
P
t
)
)
a
t
)
=
∑
t
(
δ
s
(
P
t
)
∘
s
i
)
δ
i
(
a
t
)
+
∑
t
δ
i
δ
s
(
P
t
)
a
t
.
{\displaystyle \delta _{i}\delta _{s}(f)=\sum _{t}\delta _{i}(\delta _{s}(P_{t}))a_{t})=\sum _{t}(\delta _{s}(P_{t})\circ s_{i})\delta _{i}(a_{t})+\sum _{t}\delta _{i}\delta _{s}(P_{t})a_{t}.}
Since
δ
i
δ
s
=
δ
s
i
s
{\displaystyle \delta _{i}\delta _{s}=\delta _{s_{i}s}}
or 0, it follows that
∑
t
δ
s
(
P
t
)
δ
i
(
a
t
)
∘
s
i
=
0
{\displaystyle \sum _{t}\delta _{s}(P_{t})\,\delta _{i}(a_{t})\circ s_{i}=0}
so that by the invertibility of N
δ
i
(
a
t
)
=
0
{\displaystyle \delta _{i}(a_{t})=0}
for all i, i.e. at is invariant under W.
Steinberg basis
As above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots. From these data we obtain the subset Δ = {α1, α2, …, αn} of the simple roots, the coroots
α
i
∨
=
2
(
α
i
,
α
i
)
−
1
α
i
,
{\displaystyle \alpha _{i}^{\vee }=2(\alpha _{i},\alpha _{i})^{-1}\alpha _{i},}
and the fundamental weights λ1, λ2, ..., λn as the dual basis of the coroots.
For each element s in W, let Δs be the subset of Δ consisting of the simple roots satisfying s−1α < 0, and put
λ
s
=
s
−
1
∑
α
i
∈
Δ
s
λ
i
,
{\displaystyle \lambda _{s}=s^{-1}\sum _{\alpha _{i}\in \Delta _{s}}\lambda _{i},}
where the sum is calculated in the weight lattice P.
The set of linear combinations of the exponentials eμ with integer coefficients for μ in P becomes a ring over Z isomorphic to the group algebra of P, or equivalently to the representation ring
R(T) of T, where T is a maximal torus in K, the simply connected, connected compact semisimple Lie group with root system Φ. If W is the Weyl group of Φ, then the representation ring R(K) of K can be identified with R(T)W.
Steinberg's theorem. The exponentials λs (s in W) form a free basis for the ring of exponentials over the subring of W-invariant exponentials.
Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator
A
(
ψ
)
=
∑
s
∈
W
(
−
1
)
ℓ
(
s
)
s
⋅
ψ
.
{\displaystyle A(\psi )=\sum _{s\in W}(-1)^{\ell (s)}s\cdot \psi .}
The positive roots β with sβ positive can be seen as a set of positive roots for a root system on a subspace of V; the roots are the ones orthogonal to s.λs. The corresponding Weyl group equals the stabilizer of λs in W. It is generated by the simple reflections sj for which sαj is a positive root.
Let M and N be the matrices
M
t
s
=
t
(
λ
s
)
,
N
s
t
=
(
−
1
)
ℓ
(
t
)
⋅
t
(
ψ
s
)
,
{\displaystyle M_{ts}=t(\lambda _{s}),\,\,N_{st}=(-1)^{\ell (t)}\cdot t(\psi _{s}),}
where ψs is given by the weight s−1ρ - λs. Then the matrix
B
s
,
s
′
=
Ω
−
1
(
N
M
)
s
,
s
′
=
A
(
ψ
s
λ
s
′
)
Ω
{\displaystyle B_{s,s'}=\Omega ^{-1}(NM)_{s,s'}={A(\psi _{s}\lambda _{s'}) \over \Omega }}
is triangular with respect to any total order on W such that s ≥ t implies
ℓ
(
s
)
≥
ℓ
(
t
)
{\displaystyle \ell (s)\geq \ell (t)}
.
Steinberg proved that the entries of B are W-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence its inverse C has the same form. Define
φ
s
=
∑
C
s
,
t
ψ
t
.
{\displaystyle \varphi _{s}=\sum C_{s,t}\psi _{t}.}
If χ is an arbitrary exponential sum, then it follows that
χ
=
∑
s
∈
W
a
s
λ
s
{\displaystyle \chi =\sum _{s\in W}a_{s}\lambda _{s}}
with as the W-invariant exponential sum
a
s
=
A
(
φ
s
χ
)
Ω
.
{\displaystyle a_{s}={A(\varphi _{s}\chi ) \over \Omega }.}
Indeed this is the unique solution of the system of equations
t
χ
=
∑
s
∈
W
t
(
λ
s
)
a
s
=
∑
s
M
t
,
s
a
s
.
{\displaystyle t\chi =\sum _{s\in W}t(\lambda _{s})\,\,a_{s}=\sum _{s}M_{t,s}a_{s}.}
References
Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. (1973), "Schubert cells, and the cohomology of the spaces G/P", Russian Math. Surveys, 28 (3): 1–26, doi:10.1070/RM1973v028n03ABEH001557, S2CID 250748691
Billey, Sara C. (1999), "Kostant polynomials and the cohomology ring for G/B.", Duke Math. J., 96: 205–224, CiteSeerX 10.1.1.11.8630, doi:10.1215/S0012-7094-99-09606-0, S2CID 16184223
Bourbaki, Nicolas (1981), Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, ISBN 978-2-225-76076-1
Cartan, Henri (1950), "Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie", Colloque de Topologie (Espaces Fibrés), Bruxelles: 15–27
Cartan, Henri (1950), "La transgression dans un groupe de Lie et dans un espace fibré principal", Colloque de Topologie (Espaces Fibrés), Bruxelles: 57–71
Chevalley, Claude (1955), "Invariants of finite groups generated by reflections", Amer. J. Math., 77 (4): 778–782, doi:10.2307/2372597, JSTOR 2372597
Demazure, Michel (1973), "Invariants symétriques entiers des groupes de Weyl et torsion", Invent. Math., 21 (4): 287–301, Bibcode:1973InMat..21..287D, doi:10.1007/BF01418790, S2CID 123253975
Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976), Connections, curvature, and cohomology. Volume III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, vol. 47-III, Academic Press
Humphreys, James E. (1994), Introduction to Lie Algebras and Representation Theory (2nd ed.), Springer, ISBN 978-0-387-90053-7
Kostant, Bertram (1963), "Lie algebra cohomology and generalized Schubert cells", Ann. of Math., 77 (1): 72–144, doi:10.2307/1970202, JSTOR 1970202
Kostant, Bertram (1963), "Lie group representations on polynomial rings", Amer. J. Math., 85 (3): 327–404, doi:10.2307/2373130, JSTOR 2373130
Kostant, Bertram; Kumar, Shrawan (1986), "The nil Hecke ring and cohomology of G/P for a Kac–Moody group G.", Proc. Natl. Acad. Sci. U.S.A., 83 (6): 1543–1545, Bibcode:1986PNAS...83.1543K, doi:10.1073/pnas.83.6.1543, PMC 323118, PMID 16593661
Lascoux, Alain; Schützenberger, Marcel-Paul (1982), "Polynômes de Schubert [Schubert polynomials]", Comptes Rendus de l'Académie des Sciences, Série I, 294: 447–450
McLeod, John (1979), The Kunneth formula in equivariant K-theory, Lecture Notes in Math., vol. 741, Springer, pp. 316–333
Steinberg, Robert (1975), "On a theorem of Pittie", Topology, 14 (2): 173–177, doi:10.1016/0040-9383(75)90025-7
Kata Kunci Pencarian:
- Kostant polynomial
- Bertram Kostant
- Schubert polynomial
- Harmonic polynomial
- Sara Billey
- Amitsur–Levitzki theorem
- Per Enflo
- Plancherel theorem for spherical functions
- Hochschild homology
- Capelli's identity
