Bond gcds and a-chains of saturated chains in W / W_λ #
Let λ be a dominant weight for a base of a finite reduced root datum, so that the weights and the
pairings with coroots are integers. Each cover y ⋖ x of the parabolic quotient W / W_λ has a
unique positive root label β with y λ = s_β (x λ), and a positive integer bond
b_λ(x, y) = ⟨y λ, β^∨⟩ = -⟨x λ, β^∨⟩, by TauCeti.LinearAlgebra.RootSystem.Weyl.CoverBond. For a
saturated chain C : x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l of W / W_λ this file defines
- the bond gcd
d(C), the gcd of the bondsb_λ(x_{h-1}, x_h)of its covers, with the conventiond(C) = 0for the chain of length0; - for a rational number
a, thea-chain condition of Littelmann:a ⟨x_{h-1} λ, β_h^∨⟩is an integer for every coverx_h ⋖ x_{h-1}ofCwith root labelβ_h.
The main result is the divisibility criterion: writing a = p / q in lowest terms with q > 0,
the chain C is an a-chain exactly when q divides d(C). Since p and q are coprime,
a b is an integer exactly when q ∣ b, and q divides every bond exactly when it divides their
gcd. The bond gcd of a concatenation of two chains is the gcd of their bond gcds, so being an
a-chain is checked piece by piece.
In Littelmann's description of the Lakshmibai–Seshadri paths of shape λ, a pair of cosets
τ > σ can follow each other at a rational break a exactly when there is an a-chain for
(τ, σ); the bond gcd turns that condition into a divisibility. The gcd does not depend on the
saturated chain from τ to σ, by TauCeti.bondGcd_eq_intervalBondGcd in
TauCeti.LinearAlgebra.RootSystem.Weyl.IntervalBondGcd.
The a-chain condition makes the corresponding Lakshmibai–Seshadri paths integral where it
matters. Along an a-chain from x to y every integral pairing changes by an a-integral
amount: a (f (x λ) - f (y λ)) ∈ ℤ. If moreover the chain crosses a simple wall,
⟨x λ, αᵢ^∨⟩ < 0 ≤ ⟨y λ, αᵢ^∨⟩, then its first crossing cover is the simple reflection sᵢ, by the
lifting property, and so a ⟨x λ, αᵢ^∨⟩ ∈ ℤ. For a path running along orbit weights τ_k λ
between breaks t_k linked by t_{k+1}-chains, this makes the running height
⟨π(t), αᵢ^∨⟩ integral at every break where it stops decreasing, and at the endpoint; hence its
minimum is an integer. These are the integrality properties under which Littelmann's root
operators on such paths have the string lengths ε_i = -min H_i and φ_i = H_i(1) - min H_i.
The weights live in a root datum (a root pairing over ℤ) so that bonds are integers by
construction and their gcd is defined.
Main definitions #
TauCeti.bondGcd: the gcdd(C)of the bonds of a saturated chain ofW / W_λ.TauCeti.IsAChain: thea-chain condition on a saturated chain ofW / W_λ.
Main results #
TauCeti.isAChain_iff_den_dvd_bondGcd: a saturated chain is ana-chain exactly when the denominator ofadivides its bond gcd.TauCeti.isAChain_iff_forall_coroot': thea-chain condition in Littelmann's form,a ⟨x_{h-1} λ, β_h^∨⟩ ∈ ℤfor the positive root labelβ_hof each cover.TauCeti.dvd_bondGcd_iffandTauCeti.bondGcd_dvd: the bond gcd is the gcd of the bonds, as a divisibility characterization.TauCeti.bondGcd_eq_zero_iff: the bond gcd is positive exactly when the chain is nonempty;TauCeti.bondGcd_eq_zero_of_length_eq_zerosays a chain of length0has bond gcd0.TauCeti.bondGcd_smashandTauCeti.isAChain_smash_iff: the bond gcd and thea-chain condition of a concatenation of two chains;TauCeti.bondGcd_snocandTauCeti.isAChain_snoc_ifffor the extension of a chain by one cover, andTauCeti.bondGcd_consandTauCeti.bondGcd_eq_gcd_bondGcd_tailat its start.TauCeti.bondGcd_dvd_sub: the bond gcd of a chain fromxtoydividesf (x λ) - f (y λ)for every integral linear formf.TauCeti.IsAChain.exists_mul_sub_eq_intCast: along ana-chain fromxtoy,a (f (x λ) - f (y λ))is an integer for every integral linear formf.TauCeti.IsAChain.exists_mul_coroot'_eq_intCast: ana-chain fromxtoywith⟨x λ, αᵢ^∨⟩ < 0 ≤ ⟨y λ, αᵢ^∨⟩hasa ⟨x λ, αᵢ^∨⟩ ∈ ℤ.TauCeti.exists_sum_coroot'_eq_intCast_of_isAChain,TauCeti.exists_sum_eq_intCast_of_isAChainandTauCeti.exists_isLeast_sum_coroot'_eq_intCast_of_isAChain: the heights of a path alonga-chains are integral at the breaks where they stop decreasing and at the endpoint, and their minimum is an integer.TauCeti.exists_isLeast_sum_coroot'_Icc_of_isAChain: the least height on a final piece of such a path is the height at its start or an integer.TauCeti.sum_coroot'_lt_sum_coroot'_succ_of_isAChain,TauCeti.add_one_le_sum_coroot'_of_isAChainandTauCeti.exists_strictly_rising_of_isAChain: after its last minimumh, the height of such a path rises strictly until it reachesh + 1, and it stays at leasth + 1afterwards.
References #
- P. Littelmann, A Littlewood–Richardson rule for symmetrizable Kac–Moody algebras, Invent. Math. 116 (1994), 329--346, Section 4.
- P. Littelmann, Paths and root operators in representation theory, Ann. of Math. 142 (1995), 499--525, Section 4.
- R. Chirivì, X. Fang and P. Littelmann, LS algebras, valuations and Schubert varieties, arXiv:2203.12992, Section 2.
The bond gcd d(C) of a saturated chain C : x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l of the parabolic quotient
W / W_λ: the gcd of the bonds b_λ(x_{h-1}, x_h) of its covers, where the bond of y ⋖ x is
⟨y λ, β^∨⟩ for the positive root β with y λ = s_β (x λ). The chain of length 0 has bond gcd
0.
Instances For
A saturated chain C : x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l of the parabolic quotient W / W_λ is an
a-chain, for a rational number a, when a times the bond b_λ(x_{h-1}, x_h) of each of its
covers is an integer. With the positive root label β_h of the cover x_h ⋖ x_{h-1} this is
Littelmann's condition that a ⟨x_{h-1} λ, β_h^∨⟩ is an integer
(TauCeti.isAChain_iff_forall_coroot').
Instances For
The a-chain condition is integrality of a times each reflection bond.
A natural number divides the bond gcd d(C) of a saturated chain exactly when it divides the
bond of every cover of the chain.
The bond gcd d(C) of a saturated chain divides the bond of each cover of the chain.
A saturated chain is an a-chain exactly when the denominator of a divides its bond
gcd. Writing a = p / q in lowest terms with q > 0, the integrality of a times every bond of
the chain C is the divisibility q ∣ d(C).
The chain of length 0 has bond gcd 0.
The bond gcd of the concatenation of two saturated chains, the second starting where the first ends, is the gcd of their bond gcds.
The bond gcd of a saturated chain extended by one more cover y ⋖ x_l at its end is the gcd of
its bond gcd and the bond b_λ(x_l, y) of the new cover.
The bond gcd of a saturated chain extended by one more cover x₀ ⋖ x at its start is the gcd
of the bond b_λ(x, x₀) of the new cover and the bond gcd of the chain.
The bond gcd of a saturated chain x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l of positive length is the gcd of the
bond b_λ(x₀, x₁) of its first cover and the bond gcd of its tail x₁ ⋗ ⋯ ⋗ x_l.
The concatenation of two saturated chains, the second starting where the first ends, is an
a-chain exactly when both pieces are.
The chain of length 0 is an a-chain for every a.
A saturated chain extended by one more cover y ⋖ x_l at its end is an a-chain exactly when
the chain is an a-chain and a times the bond b_λ(x_l, y) of the new cover is an integer.
An initial segment x₀ ⋗ ⋯ ⋗ x_k of an a-chain is an a-chain.
A final segment x_k ⋗ ⋯ ⋗ x_l of an a-chain is an a-chain.
The bond gcd of the chain sᵢ ⋗ 1. For a simple root αᵢ with ⟨λ, αᵢ^∨⟩ ≠ 0, the
simple reflection sᵢ covers the identity in W / W_λ, and the saturated chain sᵢ ⋗ 1 has bond
gcd |⟨λ, αᵢ^∨⟩|. So it is an a-chain exactly when the denominator of a divides
⟨λ, αᵢ^∨⟩.
The a-chain condition in Littelmann's form. A saturated chain x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l of
W / W_λ is an a-chain exactly when a ⟨x_{h-1} λ, β^∨⟩ is an integer for every cover
x_h ⋖ x_{h-1} and every positive root β with x_h λ = s_β (x_{h-1} λ). Such a root exists and
is unique (TauCeti.existsUnique_isPos_reflection_smul_eq_of_minimalCosetRepCovBy), and
⟨x_{h-1} λ, β^∨⟩ < 0.
A saturated chain of W / W_λ of length 0 has bond gcd 0, the gcd of no bonds.
The bond gcd of a saturated chain of W / W_λ vanishes exactly when the chain has length 0:
the bonds of covers are positive.
The bond gcd divides the change of every integral linear form along the chain. If
C : x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l is a saturated chain of W / W_λ, then d(C) divides
f (x₀ λ) - f (x_l λ) for every integral linear form f on the weight lattice: each cover changes
the weight by its bond times a root.
Integrality along a-chains #
Integral linear forms on the two ends of an a-chain differ by an a-integral amount. If
C : x₀ ⋗ x₁ ⋗ ⋯ ⋗ x_l is an a-chain of W / W_λ, then a (f (x₀ λ) - f (x_l λ)) is an
integer for every integral linear form f on the weight lattice, for instance for the pairing
⟨-, β^∨⟩ with any coroot.
Along an a-chain, the scaled difference of its endpoint weights is an actual root-lattice
vector, in every characteristic-zero linear realization. This retains the lattice information,
not only its values under integral linear forms.
Littelmann's integrality at a simple wall. Let C : x₀ ⋗ ⋯ ⋗ x_l be an a-chain of
W / W_λ and αᵢ a simple root with ⟨x₀ λ, αᵢ^∨⟩ < 0 ≤ ⟨x_l λ, αᵢ^∨⟩. Then a ⟨x₀ λ, αᵢ^∨⟩
is an integer. At the first cover x_h ⋖ x_{h-1} crossing the wall, the root label is αᵢ
(TauCeti.reflection_smul_eq_of_minimalCosetRepCovBy_of_coroot'_neg), so a ⟨x_{h-1} λ, αᵢ^∨⟩
is an integer, and the part of C before it changes the pairing by an a-integral amount.
Heights of Lakshmibai–Seshadri paths at their breaks #
A Lakshmibai–Seshadri path of shape λ runs, for 0 ≤ k < s, along the orbit weight τ_k λ
during the time interval [t_k, t_{k+1}], starting at time t_0 = 0, and each break t_{k+1}
between τ_k and τ_{k+1} carries a t_{k+1}-chain from τ_k down to τ_{k+1}. Its value at
the j-th break, paired with an integral linear form f, is the vertex height
∑_{k < j} (t_{k+1} - t_k) f (τ_k λ). The results below need only the chains, not the order of
the cosets or of the times, except where stated.
Heights at breaks, up to integers. If consecutive orbit weights τ_k λ and τ_{k+1} λ are
joined by t_{k+1}-chains for k + 1 < s and t_0 = 0, then for j < s the height
∑_{k ≤ j} (t_{k+1} - t_k) f (τ_k λ) at the break t_{j+1} differs from t_{j+1} f (τ_j λ) by
an integer, for every integral linear form f.
Littelmann's integrality of Lakshmibai–Seshadri paths at local minima. With the chains of
TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, let αᵢ be a simple root such that the
height ⟨-, αᵢ^∨⟩ stops decreasing at the break t_{j+1}: ⟨τ_j λ, αᵢ^∨⟩ < 0 ≤ ⟨τ_{j+1} λ, αᵢ^∨⟩. Then the height ∑_{k ≤ j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩ there is an
integer.
The endpoint of a Lakshmibai–Seshadri path is integral. With the chains of
TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, if the final time t_s is an integer (for a
Lakshmibai–Seshadri path, t_s = 1), then the final height ∑_{k < s} (t_{k+1} - t_k) f (τ_k λ)
is an integer for every integral linear form f.
Least heights of a Lakshmibai–Seshadri path on a final piece. With the chains of
TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, suppose the times increase strictly,
t_0 < t_1 < ⋯ < t_s, and the final time t_s is an integer. Then for each simple root αᵢ and
each m ≤ s, the least vertex height ∑_{k < j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩, m ≤ j ≤ s, is
either the height at m or an integer: away from m, it is attained at a break where the height
stops decreasing or at the endpoint.
The minimal height of a Lakshmibai–Seshadri path is integral. With the chains of
TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, suppose the times increase strictly,
t_0 < t_1 < ⋯ < t_s, and the final time t_s is an integer. Then for each simple root αᵢ
the least vertex height ∑_{k < j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩, j ≤ s, is an integer. Since
the height is affine between breaks, this least vertex height is the minimum of the height along
the whole path.
The height of a Lakshmibai–Seshadri path after a minimum #
Littelmann's lowering operator f_αᵢ reflects the piece of a path between the last time t₀ at
which the height ⟨π(t), αᵢ^∨⟩ is minimal and the first later time t₁ at which it has risen by
one. For a Lakshmibai–Seshadri path this piece is as simple as possible: the height rises
strictly from t₀ to t₁, and it stays at least the minimum plus one after t₁. Both statements
hold more generally after any vertex at which the height is an integer, provided the height stays
strictly above that integer afterwards.
The height of a Lakshmibai–Seshadri path rises strictly after a minimum, up to the next
integer. With the chains of TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, strictly
increasing times and an integral final time, write
H j = ∑_{k < j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩. Suppose that the height H k₀ is an integer and
that H k > H k₀ for k₀ < k ≤ s. Then H j < H (j + 1) for every k₀ ≤ j < s with
H j < H k₀ + 1: a stationary height would be integral, and a decreasing one would lead to a
local minimum strictly between the integers H k₀ and H k₀ + 1.
The height of a Lakshmibai–Seshadri path stays up after rising by one from a minimum. With
the chains of TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, strictly increasing times and an
integral final time, write H j = ∑_{k < j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩. Suppose that the height
H k₀ is an integer and that H k > H k₀ for k₀ < k ≤ s. If H k₁ ≥ H k₀ + 1 for some
k₁ > k₀, then H j ≥ H k₀ + 1 for every k₁ ≤ j ≤ s: the least height after k₁ is attained at
k₁ or is an integer greater than H k₀.
The rising piece of a Lakshmibai–Seshadri path at a lowering step. With the chains of
TauCeti.exists_sum_sub_mul_eq_intCast_of_isAChain, strictly increasing times and an integral final
time, write H j = ∑_{k < j} (t_{k+1} - t_k) ⟨τ_k λ, αᵢ^∨⟩ and suppose that the final height H s
exceeds some height H j, j ≤ s, by at least one. Then there are vertices k₀ < k₁ ≤ s such
that H k₀ is the minimal height, an integer, the height rises strictly from k₀ to k₁ while
staying below H k₀ + 1 before k₁, and it is at least H k₀ + 1 from k₁ on. One may take
for k₀ the last vertex of minimal height and for k₁ the first later vertex of height at least
H k₀ + 1.