The Weyl orbit of a dominant weight as a parabolic quotient #
Let λ be a dominant weight for a base b of a finite crystallographic reduced root system. Its
stabilizer W_λ in the Weyl group is the standard parabolic subgroup of the Coxeter system
TauCeti.weylCoxeterSystem generated by the simple reflections in the walls through λ. So the
orbit W λ is the parabolic quotient W / W_λ, and each orbit weight u λ has a unique minimal
coset representative. This file records that dictionary and the reflection test comparing two
orbit weights related by a reflection.
For a positive root β and an orbit weight u λ, let u and v be the minimal representatives
of the cosets of u and of s_β u, so that v λ = s_β (u λ). Then, in the Bruhat order,
v < uwhen⟨u λ, β^∨⟩ < 0;u < vwhen⟨u λ, β^∨⟩ > 0;v = uwhen⟨u λ, β^∨⟩ = 0, and then alsos_β (u λ) = u λ.
The comparison is between the minimal representatives, not between arbitrary Weyl-group elements
with the given weights: on a chamber wall W_λ is nontrivial, and s_β u itself is generally not
minimal in its coset. A weak inequality gives no strict comparison: on a wall the zero pairing
gives the same coset.
This is the order test on W / W_λ underlying the description of Lakshmibai–Seshadri paths by
saturated chains of orbit weights, in which each step is a reflection ξ ↦ s_β ξ with
⟨ξ, β^∨⟩ < 0.
Main definitions #
RootPairing.wallIndices: the simple roots whose coroot vanishes on a weight, as a set of simple generators ofTauCeti.weylCoxeterSystem.
Main results #
RootPairing.parabolic_wallIndices_eq_stabilizer: the standard parabolic subgroup of the walls through a dominant weight is its stabilizer.TauCeti.inv_mul_mem_parabolic_wallIndices_iff: two Weyl-group elements lie in the same coset of that parabolic subgroup exactly when they move the dominant weight to the same point.TauCeti.existsUnique_isMinimalCosetRep_smul_eq: each orbit weight has a unique minimal coset representative.TauCeti.bruhatLE_of_isMinimalCosetRep_of_smul_eq: that representative lies below every Weyl-group element with the same orbit weight.TauCeti.bruhatStep_ofIdx_mul_of_coroot'_neg:s_β w < win the Weyl group itself when⟨w λ, β^∨⟩ < 0.TauCeti.coroot'_smul_nonneg_of_not_isLeftDescentandTauCeti.coroot'_smul_nonpos_of_isLeftDescent: the sign of⟨w λ, αᵢ^∨⟩for a simple rootαᵢaccording to whethersᵢis a left descent ofw.TauCeti.bruhatLE_and_ne_of_isMinimalCosetRep_of_coroot'_neg,TauCeti.bruhatLE_and_ne_of_isMinimalCosetRep_of_coroot'_posandTauCeti.eq_of_isMinimalCosetRep_of_coroot'_eq_zero: the reflection test in its three cases.TauCeti.bruhatLE_and_ne_iff_coroot'_neg_of_isMinimalCosetRepandTauCeti.eq_iff_coroot'_eq_zero_of_isMinimalCosetRep: the reflection test as characterizations of the strict order and of equality.TauCeti.reflection_smul_eq_of_minimalCosetRepCovBy_of_coroot'_neg: a covery ⋖ xwith⟨x λ, αᵢ^∨⟩ < 0 ≤ ⟨y λ, αᵢ^∨⟩is given by the simple reflection,y λ = sᵢ (x λ).TauCeti.simple_mul_smul_eq_reflectionandTauCeti.coroot'_simple_mul_smul_eq_neg: left multiplication bysᵢreflects orbit weights inαᵢ, negating the pairing withαᵢ^∨.TauCeti.isMinimalCosetRep_simple_mul_of_coroot'_ne_zeroandTauCeti.minimalCosetRepCovBy_simple_mul_of_coroot'_pos: off the wall ofαᵢ,sᵢ uis again minimal, andu ⋖ sᵢ uwhen⟨u λ, αᵢ^∨⟩ > 0.TauCeti.minimalCosetRepCovBy_simple_mul_simple_mul_of_coroot'_posandTauCeti.minimalCosetRepCovBy_simple_mul_simple_mul_of_coroot'_neg:sᵢcarries covers on one side of its wall to covers.TauCeti.eq_simple_mul_of_minimalCosetRepCovBy_of_coroot'_nonpos: a covery ⋖ xwith⟨x λ, αᵢ^∨⟩ ≤ 0 < ⟨y λ, αᵢ^∨⟩isx = sᵢ y.
References #
- A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Springer GTM 231 (2005), Sections 2.4 and 2.5.
- J. E. Humphreys, Reflection Groups and Coxeter Groups, CUP (1990), Sections 1.10 and 5.7.
- P. Littelmann, Paths and root operators in representation theory, Ann. of Math. 142 (1995), 499--525, Section 4.
The walls through a weight as simple generators #
The simple roots of the base b whose simple coroot vanishes on x, as a set of simple
generators of TauCeti.weylCoxeterSystem. For dominant x the standard parabolic subgroup they
generate is the stabilizer of x (RootPairing.parabolic_wallIndices_eq_stabilizer).
Instances For
A simple root lies in RootPairing.wallIndices exactly when its coroot vanishes on the
weight.
A nonzero scalar multiple of a weight lies on exactly the same simple walls.
A nonzero natural multiple of a weight lies on exactly the same simple walls.
Identify minimal wall-parabolic representatives before and after nonzero natural scaling. The underlying Weyl-group element is unchanged.
Instances For
Scaling the shape preserves the Weyl-group element of a minimal representative.
Inverse scaling preserves the Weyl-group element of a minimal representative.
Adding a weight χ killed by the simple coroots, such as a central weight, does not change
the walls through a weight.
The stabilizer of a dominant weight is a standard parabolic subgroup: the one generated by the simple reflections in the walls through it.
Cosets of the stabilizer are orbit weights. Two Weyl-group elements lie in the same left coset of the standard parabolic subgroup of the walls through a dominant weight exactly when they move that weight to the same point.
Each orbit weight of a dominant weight has a unique minimal coset representative.
Minimal coset representatives are determined by their orbit weight.
The minimal coset representative of an orbit weight is Bruhat-below every Weyl-group element with that orbit weight.
The reflection test #
Reflection test in the Weyl group. Reflecting in a positive root β with
⟨w λ, β^∨⟩ < 0 moves w one step down the Bruhat graph: s_β w < w, since the root w⁻¹ β
pairs negatively with the dominant weight λ and so is a negative root.
A simple reflection that does not shorten w pairs nonnegatively. If sᵢ is not a left
descent of w, then ⟨w λ, αᵢ^∨⟩ ≥ 0 for every dominant λ.
A left descent pairs nonpositively. If sᵢ is a left descent of w, then
⟨w λ, αᵢ^∨⟩ ≤ 0 for every dominant λ.
Reflection test, negative pairing. Let β be a positive root with ⟨w λ, β^∨⟩ < 0, and
let v be the minimal coset representative of s_β w, so v λ = s_β (w λ). Then v < w in the
Bruhat order. Here w is arbitrary; it need not be minimal in its own coset.
Reflection test, positive pairing. Let u be a minimal coset representative and β a
positive root with ⟨u λ, β^∨⟩ > 0. Then u < w in the Bruhat order for every w with
w λ = s_β (u λ), in particular for the minimal coset representative of s_β u.
Reflection test, zero pairing. If ⟨u λ, β^∨⟩ = 0, then s_β fixes u λ, so u and
s_β u have the same minimal coset representative.
The reflection test characterizes the strict Bruhat order on minimal coset
representatives. If u and v are the minimal coset representatives of u and s_β u for a
positive root β, then v < u exactly when ⟨u λ, β^∨⟩ < 0.
The reflection test characterizes equal cosets. If u and v are the minimal coset
representatives of u and s_β u for a root β, then v = u exactly when ⟨u λ, β^∨⟩ = 0.
Covers crossing a simple wall #
A cover leaving the negative side of a simple wall is the simple reflection. If y ⋖ x is
a cover of W / W_λ and αᵢ is a simple root with ⟨x λ, αᵢ^∨⟩ < 0 ≤ ⟨y λ, αᵢ^∨⟩, then
y λ = sᵢ (x λ). By the reflection test the minimal representative v of sᵢ x lies strictly
below x; since sᵢ does not shorten y, the lifting property gives y ≤ v, and the cover
forces y = v.
Left multiplication by a simple reflection #
Left multiplication by the simple reflection sᵢ acts on orbit weights as the reflection in
αᵢ.
⟨(sᵢ u) λ, αᵢ^∨⟩ = -⟨u λ, αᵢ^∨⟩.
A simple reflection off the wall keeps a minimal coset representative minimal. If u is a
minimal coset representative of W / W_λ and ⟨u λ, αᵢ^∨⟩ ≠ 0, then sᵢ u is a minimal coset
representative too.
A simple reflection on the positive side of its wall covers. If u is a minimal coset
representative of W / W_λ with ⟨u λ, αᵢ^∨⟩ > 0, then u ⋖ sᵢ u. Its bond is ⟨u λ, αᵢ^∨⟩, by
TauCeti.reflectionBond_simple_mul_smul_smul.
sᵢ moves covers on the positive side of its wall up. If y ⋖ x in W / W_λ with
⟨x λ, αᵢ^∨⟩ > 0 and ⟨y λ, αᵢ^∨⟩ > 0, then sᵢ y ⋖ sᵢ x.
sᵢ moves covers on the negative side of its wall down. If y ⋖ x in W / W_λ with
⟨x λ, αᵢ^∨⟩ < 0 and ⟨y λ, αᵢ^∨⟩ < 0, then sᵢ y ⋖ sᵢ x.
A cover leaving the nonpositive side of a simple wall is given by the simple reflection. If
y ⋖ x in W / W_λ with ⟨x λ, αᵢ^∨⟩ ≤ 0 < ⟨y λ, αᵢ^∨⟩, then x = sᵢ y.