Lakshmibai-Seshadri paths for hyperbolic Kac-Moody algebras of rank 2
aa r X i v : . [ m a t h . QA ] A ug Lakshmibai-Seshadri paths forhyperbolic Kac-Moody algebras of rank 2
Dongxiao Yu
Graduate School of Pure and Applied Sciences, University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan(e-mail: [email protected])
Abstract
Let g be a hyperbolic Kac-Moody algebra of rank 2, and set λ := Λ − Λ ,where Λ , Λ are the fundamental weights for g ; note that λ is neither dominantnor antidominant. Let B ( λ ) be the crystal of all Lakshmibai-Seshadri paths ofshape λ . We prove that (the crystal graph of) B ( λ ) is connected. Furthermore,we give an explicit description of Lakshmibai-Seshadri paths of shape λ . Let g be a symmetrizable Kac-Moody algebra over C with h the Cartan subalgebra.We denote by W the Weyl group of g . Let P be an integral weight lattice of g , P + theset of dominant integral weights, and − P + the set of antidominant weights. In [L1, L2],Littelmann introduced the notion of Lakshmibai-Seshadri (LS for short) paths of shape λ ∈ P , and gave the set B ( λ ) of all LS paths of shape λ a crystal structure. Kashiwara[Ka3] and Joseph [J] proved independently that if λ ∈ P + (resp., λ ∈ − P + ), then B ( λ )is isomorphic, as a crystal, to the crystal basis of the integrable highest (resp., lowest)weight module of highest weight (resp., lowest weight) λ . Since B ( λ ) = B ( wλ ) for every λ ∈ P and w ∈ W by the definition of LS paths, we can easily see that if λ ∈ P satisfies W λ ∩ P + = ∅ (resp., W λ ∩ ( − P + ) = ∅ ), then B ( λ ) is isomorphic, as a crystal, to thecrystal basis of an integrable highest (resp., lowest) module. Here are natural questions:How are the crystal structure of B ( λ ) and its relation to the representation theory inthe case that λ ∈ P satisfies W λ ∩ ( P + ∪ ( − P + )) = ∅ ?If g is of finite type, then there is no λ ∈ P such that W λ ∩ ( P + ∪ ( − P + )) = ∅ ; it iswell-known that W λ ∩ P + = ∅ for any λ ∈ P . Assume that g is of affine type, and let c ∈ h be the canonical central element of g . Then, W λ ∩ ( P + ∪ ( − P + )) = ∅ if and onlyif ( λ = 0, and) h λ, c i = 0. Naito and Sagaki proved in [NS1] and [NS2] that if λ is of theform: λ = m̟ i , where m ∈ Z ≥ and ̟ i is the level-zero fundamental weight (note that h ̟ i , c i = 0), then B ( m̟ i ) is isomorphic, as a crystal, to the crystal basis B ( m̟ i ) of the1xtremal weight module of extremal weight m̟ i over the quantum affine algebra U q ( g ).Here, (for an arbitrary symmetrizable Kac-Moody algebra g and an arbitrary integralweight λ ∈ P for g ,) the extremal weight module of extremal weight λ is the integrable U q ( g )-module generated by a single element v λ with the defining relation that “ v λ is anextremal weight vector of weight λ ”; this module was introduced by Kashiwara [Ka1, §
8] as a natural generalization of integrable highest (or lowest) weight modules, and hasa crystal basis B ( λ ) ([Ka1, § B ( λ ) for general λ ∈ P such that h λ, c i = 0, and in [INS], they proved that thereexists a canonical surjective (not bijective in general) strict morphism of crystals from B ( λ ) onto B ( λ ).So, in this paper, we consider the case where g = g ( A ) is a hyperbolic Kac-Moodyalgebra of rank 2, associated to the generalized Cartan matrix A = (cid:18) − a − b (cid:19) ( a, b ∈ Z ≥ , ab > . Let Λ , Λ be the fundamental weights for g , and set λ := Λ − Λ . In Proposition3.1.1, we prove that W λ ∩ ( P + ∪ ( − P + )) = ∅ if a, b ≥
2; in fact, if a = 1 (resp., b = 1),then W λ ∩ P + = ∅ (resp., W λ ∩ ( − P + ) = ∅ ); see Remark 3.1.2. Then we prove thefollowing theorem. Theorem 1.
The crystal graph of B (Λ − Λ ) is connected.Our weight λ = Λ − Λ can be considered as an analog of the level-zero fundamentalweight for a (rank 2) affine Lie algebra. So, in a future work, we will study, as in theaffine case, the relation between B (Λ − Λ ) and the crystal basis B (Λ − Λ ) of theextremal weight module of extremal weight Λ − Λ . In (the proof of B ( λ ) ∼ = B ( λ ) for λ ∈ P + in [Ka3] and [J], and) the proof of B ( ̟ i ) ∼ = B ( ̟ i ) in [NS1] and [NS2], theconnectedness of these crystals played very important roles. Therefore, Theorem 1 willbe strongly related to the representation theory.Finally, in the case where a, b ≥
2, we give an explicit description of LS paths ofshape Λ − Λ . Theorem 2.
Assume that a, b ≥
2. An LS path of shape λ = Λ − Λ is of the form(i) or (ii):(i) ( x m + s − λ, . . . , x m +1 λ, x m λ ; σ , σ , . . . , σ s ), where m ≥ , s ≥
1, and 0 = σ < σ < · · · < σ s = 1 satisfy the condition that p m + s − u σ u ∈ Z for 1 ≤ u ≤ s − y m − s +1 λ, . . . , y m − λ, y m λ ; δ , δ , . . . , δ s ), where m ≥ s − , s ≥
1, and 0 = δ < δ < · · · < δ s = 1 satisfy the condition that q m − s + u +1 δ u ∈ Z for 1 ≤ u ≤ s − x m , y m ∈ W, m ≥ , are defined in (2.2), (2.3), and the sequences { p m } m ≥ and { q m } m ≥ are defined in (3.1), (3.2).This paper is organized as follows. In Section 2, we fix our notation, and recallthe definitions and several properties of LS paths. In Section 3, after showing somelemmas, we give a proof of Theorem 1. In Section 4, we give the explicit description ofLS paths of shape λ = Λ − Λ (Theorem 2), after showing some technical lemmas.2 Preliminaries.
Let A = (cid:18) − a − b (cid:19) , where a, b ∈ Z > and ab > , (2.1)be a hyperbolic generalized Cartan matrix of rank 2. Let g = g ( A ) be the Kac-Moody algebra associated to A over C . Denote by h the Cartan subalgebra of g , { α , α } ⊂ h ∗ := Hom C ( h , C ) the set of simple roots, and { α ∨ , α ∨ } ⊂ h the set ofsimple coroots; we set I = { , } . We denote by W = h r , r i the Weyl group of g ,where r i is the simple reflection in α i for i = 1 ,
2; note that W = { x m , y m | m ∈ Z ≥ } ,where x m := ( ( r r ) k if m = 2 k with k ∈ Z ≥ ,r ( r r ) k if m = 2 k + 1 with k ∈ Z ≥ . (2.2) y m := ( ( r r ) k if m = 2 k with k ∈ Z ≥ ,r ( r r ) k if m = 2 k + 1 with k ∈ Z ≥ . (2.3)Let ∆ +re denote the set of positive real roots. We see that∆ +re = (cid:8) x l ( α ) , y l +1 ( α ) | l ∈ Z even ≥ (cid:9) ⊔ (cid:8) y l ( α ) , x l +1 ( α ) | l ∈ Z even ≥ (cid:9) , (2.4)where Z even ≥ denotes the set of even nonnegative integers. Remark 2.1.1.
In fact, we know from [Kac, 5.25] that∆ +re = { c j α + d j +1 α and c j +1 α + d j α | j ≥ } , where the sequences { c j } j ≥ and { d j } j ≥ are defined by c = d = 0 , d = c = 1 , and ( c j +2 = ad j +1 − c j ,d j +2 = bc j +1 − d j . For a positive real root β ∈ ∆ +re , we denote by β ∨ the dual root of β , and by r β ∈ W the reflection in β .Let Λ , Λ ∈ h ∗ be the fundamental weights for g , i.e., h Λ i , α ∨ j i = δ i,j for i, j = 1 , P := Z Λ ⊕ Z Λ . Let P + := Z ≥ Λ + Z ≥ Λ ⊂ P be the set of dominantintegral weights, and − P + the set of antidominant integral weights.3 .2 Lakshmibai-Seshadri paths. Let us recall the definition of Lakshmibai-Seshadri paths from [L2, § Definition 2.2.1.
Let λ ∈ P be an integral weight. For µ, ν ∈ W λ , we write µ ≥ ν if there exist a sequence µ = µ , µ , . . . , µ r = ν of elements in W λ and a sequence β , β , . . . , β r of positive real roots such that µ k = r β k ( µ k − ) and h µ k − , β ∨ k i < k = 1 , , . . . , r . The sequence µ , µ , . . . , µ r above is called a chain for ( µ, ν ). If µ ≥ ν , then we define dist( µ, ν ) to be the maximal length r of all possible chains for( µ, ν ). Remark 2.2.2.
Let µ, ν ∈ W λ be such that µ > ν with dist( µ, ν ) = 1. Then thereexists a unique β ∈ ∆ +re such that r β ( µ ) = ν .Let λ ∈ P . The Hasse diagram of W λ is, by definition, the ∆ +re -labeled, directedgraph with vertex set
W λ , and edges of the following form: ν β −→ µ for µ, ν ∈ W λ and β ∈ ∆ +re such that µ > ν with dist( µ, ν ) = 1 and ν = r β ( µ ). Definition 2.2.3.
Let λ ∈ P, µ, ν ∈ W λ with µ > ν , and 0 < σ < σ -chain for ( µ, ν ) is, by definition, a decreasing sequence µ = µ > µ > · · · >µ r = ν of elements in W λ such that dist( µ k − , µ k ) = 1 and σ h µ k − , β ∨ k i ∈ Z < for all k = 1 , , . . . , r , where β k is the unique positive real root such that µ k = r β k ( µ k − ). Definition 2.2.4.
Let λ ∈ P . Let ( ν ; σ ) be a pair of a sequence ν : ν > ν > · · · > ν s of elements in W λ and a sequence σ : 0 = σ < σ < · · · < σ s = 1 of rational numbers,where s ≥
1. The pair ( ν ; σ ) is called a Lakshmibai-Seshadri (LS for short) path ofshape λ , if for each k = 1 , , . . . , s −
1, there exists a σ k -chain for ( ν k , ν k +1 ). We denoteby B ( λ ) the set of all LS paths of shape λ .We identify π = ( ν , ν , . . . , ν s ; σ , σ , . . . , σ s ) ∈ B ( λ ) with the followingpiecewise-linear continuous map π : [0 , → R ⊗ Z P : π ( t ) = j − X k =1 ( σ k − σ k − ) ν k + ( t − σ j − ) ν j for σ j − ≤ t ≤ σ j , ≤ j ≤ s. Now, we endow B ( λ ) with a crystal structure as follows ; for the axiom of crystal,see [HK, Definition 4.5.1]. First, we define wt( π ) := π (1) for π ∈ B ( λ ); we know from[L2, Lemma 4.5] that π (1) ∈ P . Next, for π ∈ B ( λ ) and i ∈ I , we define H πi ( t ) := h π ( t ) , α ∨ i i for t ∈ [0 , , m πi := min { H πi ( t ) | t ∈ [0 , } . (2.5)We know from [L2, § H πi ( t ) are integers; (2.6)4n particular, m πi is a nonpositive integer, and H πi (1) − m πi is a nonnegative integer. Wedefine e i π as follows: If m πi = 0, then we set e i π := , where is an extra element notcontained in any crystal. If m πi ≤ −
1, then we set t := min { t ∈ [0 , | H πi ( t ) = m πi } ,t := max { t ∈ [0 , t ] | H πi ( t ) = m πi + 1 } ; (2.7)by (2.6), we see that H πi ( t ) is strictly decreasing on [ t , t ] . (2.8)We define ( e i π )( t ) := π ( t ) if 0 ≤ t ≤ t ,r i ( π ( t ) − π ( t )) + π ( t ) if t ≤ t ≤ t ,π ( t ) + α i if t ≤ t ≤ §
4] that e i π ∈ B ( λ ).Similarly, we define f i π as follows : If H πi (1) − m πi = 0, then we set f i π := . If H πi (1) − m πi ≥
1, then we set t := max { t ∈ [0 , | H πi ( t ) = m πi } ,t := min { t ∈ [ t , | H πi ( t ) = m πi + 1 } ; (2.9)by (2.6), we see that H πi ( t ) is strictly increasing on [ t , t ] . (2.10)We define ( f i π )( t ) := π ( t ) if 0 ≤ t ≤ t ,r i ( π ( t ) − π ( t )) + π ( t ) if t ≤ t ≤ t ,π ( t ) − α i if t ≤ t ≤ §
4] that f i π ∈ B ( λ ). We set e i = f i := for i ∈ I .Finally, for π ∈ B ( λ ) and i ∈ I , we set ε i ( π ) := max { n ∈ Z ≥ | e ni π = } ,ϕ i ( π ) := max { n ∈ Z ≥ | f ni π = } . Theorem 2.2.5 ([L2, § § . The set B ( λ ), together with the maps wt : B ( λ ) → P, e i , f i : B ( λ ) ∪ { } → B ( λ ) ∪ { } , i ∈ I, and ε i , ϕ i : B ( λ ) → Z ≥ , i ∈ I , becomes acrystal.For π = ( ν , ν , . . . , ν s ; σ , σ , . . . , σ s ) ∈ B ( λ ), we set ι ( π ) := ν and κ ( π ) := ν s . For π ∈ B ( λ ) and i ∈ I , we set f max i π := f ϕ i ( π ) i π and e max i π := e ε i ( π ) i π . Lemma 2.2.6 ([L1, Proposition 4.2], [L2, Proposition 4.7]) . Let π ∈ B ( λ ), and i ∈ I . If h κ ( π ) , α ∨ i i >
0, then κ ( f max i ( π )) = r i κ ( π ). If h ι ( π ) , α ∨ i i <
0, then ι ( e max i ( π )) = r i ι ( π ).5 Connectedness of the crystal B (Λ − Λ ) . P + nor − P + . If λ ∈ P + (resp., λ ∈ − P + ), then B ( λ ) is isomorphic, as a crystal, to the crystalbasis of the integrable highest (resp., lowest) weight module of highest (resp., lowest)weight λ (see Kashiwara [Ka1] and Joseph [J]). Also, we see by the definition of LSpaths that B ( wλ ) = B ( λ ) for every λ ∈ P and w ∈ W . Hence, if λ ∈ P satisfies W λ ∩ P + = ∅ (resp., W λ ∩ ( − P + ) = ∅ ), then B ( λ ) is isomorphic to the crystal basis ofan integrable highest (resp., lowest) weight module. So, we focus on the case that λ ∈ P satisfies W λ ∩ ( P + ∪ ( − P + )) = ∅ . The following proposition gives a “fundamental”example of such λ . Proposition 3.1.1.
Assume that a, b = 1 in (2.1). If λ = Λ − Λ , then W λ ∩ ( P + ∪ ( − P + )) = ∅ . Remark 3.1.2. If a = 1, then we have y (Λ − Λ ) = r (Λ − Λ ) = Λ ∈ P + . If b = 1,then we have x (Λ − Λ ) = r (Λ − Λ ) = − Λ ∈ − P + .Keep the setting in Proposition 3.1.1. We define { p m } m ∈ Z ≥ and { q m } m ∈ Z ≥ by: p = p = 1 and p m +2 = ( bp m +1 − p m (if m is even) ,ap m +1 − p m (if m is odd) , (3.1) q = q = 1 and q m +2 = ( aq m +1 − q m (if m is even) ,bq m +1 − q m (if m is odd) . (3.2)Then we see that 1 = p = p ≤ p < p < · · · and 1 = q = q ≤ q < q < · · · ; notethat p = p if and only if b = 2, and q = q if and only if a = 2. Proposition 3.1.1follows immediately from the following lemmas and the fact that W = { x m , y m | m ∈ Z ≥ } (see (2.2) and (2.3)). Lemma 3.1.3.
Keep the setting in Proposition 3.1.1. For m ∈ Z ≥ , x m λ = ( p m +1 Λ − p m Λ if m is even , − p m Λ + p m +1 Λ if m is odd , (3.3) y m λ = ( q m Λ − q m +1 Λ if m is even , − q m +1 Λ + q m Λ if m is odd . (3.4) Proof.
We give a proof only for (3.3); the proof for (3.4) is similar. We show (3.3) byinduction on m . If m = 0 or m = 1, then (3.3) is obvious. Assume that m >
1. If m iseven, then x m +1 λ = r ( x m λ ) = r ( p m +1 Λ − p m Λ ) = − p m +1 Λ + ( bp m +1 − p m )Λ . m is even, we have bp m +1 − p m = p m +2 by the definition (3.1). Therefore, weobtain x m +1 λ = − p m +1 Λ + p m +2 Λ , as desired. If m is odd, then x m +1 λ = r ( x m λ ) = r ( − p m Λ + p m +1 Λ ) = ( ap m +1 − p m )Λ − p m +1 Λ . Since m is odd, we have ap m +1 − p m = p m +2 by the definition (3.2). Therefore, weobtain x m +1 λ = p m +2 Λ − p m +1 Λ , as desired. Theorem 3.2.1.
The crystal graph of B (Λ − Λ ) is connected.If a = 1 or b = 1, then B (Λ − Λ ) is connected by Remark 3.1.2, together withthe argument preceding Proposition 3.1.1. Therefore, in what follows, we assume that a, b = 1. In order to prove Theorem 3.2.1 in this case, we need some lemmas; we set λ = Λ − Λ . Lemma 3.2.2.
Let m ∈ Z ≥ , and β ∈ ∆ +re .(1) Assume that m is even. Then, h x m λ, β ∨ i ∈ Z < if and only if β = x l ( α ) or y l +1 ( α )for some l ∈ Z even ≥ .(2) Assume that m is odd. Then, h x m λ, β ∨ i ∈ Z < if and only if β = y l ( α ) or x l +1 ( α )for some l ∈ Z even ≥ . Proof.
We give a proof only for part (1); the proof for part (2) is similar. First we showthe “if” part of part (1). Let l ∈ Z even ≥ . We have h x m λ, x l ( α ∨ ) i = h x − l x m λ, α ∨ i .Here, if m ≥ l (resp., m ≤ l ), then x − l x m is equal to x m − l (resp., y l − m ). Therefore, by(3.3), we have h x m λ, x l ( α ∨ ) i = − p m − l ∈ Z < (resp., = − q l − m +1 ∈ Z < ). Similarly, wecan show that h x m λ, y l +1 ( α ∨ ) i < β = x l +1 ( α ) or y l ( α ) for l ∈ Z even ≥ , then h x m λ, β ∨ i >
0. We have h x m λ, x l +1 ( α ∨ ) i = h x − l +1 x m λ, α ∨ i = h x m + l +1 λ, α ∨ i . By (3.3), we have h x m λ, x l +1 ( α ∨ ) i = p m + l +2 > h x m λ, y l ( α ∨ ) i >
0. This completes the proof of the lemma.The next lemma can be shown in exactly the same way as Lemma 3.2.2.
Lemma 3.2.3.
Let m ∈ Z ≥ and β ∈ ∆ +re .(1) Assume that m is even. Then, h y m λ, β ∨ i ∈ Z < if and only if β = x l ( α ) or y l +1 ( α )for some l ∈ Z even ≥ .(2) Assume that m is odd. Then, h y m λ, β ∨ i ∈ Z < if and only if β = y l ( α ) or x l +1 ( α )for some l ∈ Z even ≥ . Lemma 3.2.4. (1) For m ∈ Z ≥ , we have x m λ > x m − λ with dist( x m λ, x m − λ ) = 1.And r i x m λ = x m − λ , where i = ( m is even) , m is odd) .
72) For m ∈ Z ≥ , we have y m − λ > y m λ with dist( y m − λ, y m λ ) = 1. And r j y m λ = y m − λ , where j = ( m is even) , m is odd) . Proof.
We give a proof only for part (1); the proof for part (2) is similar. We see fromLemma 3.2.2 that h x m λ, α ∨ i i <
0. Therefore, we obtain that x m λ > r i x m λ = x m − λ .Since h x m − λ, α ∨ i i >
0, we see by [L2, § r i x m λ, x m − λ ) =dist( x m λ, x m − λ ) −
1. Since dist( r i x m λ, x m − λ ) = dist( x m − λ, x m − λ ) = 0, we obtaindist( x m λ, x m − λ ) = 1, as desired. Proposition 3.2.5.
The Hasse diagram of
W λ is · · · α ←− x λ α ←− x λ α ←− x λ = λ = y λ α ←− y λ α ←− y λ α ←− · · · . Proof.
Let µ, ν ∈ W λ be such that µ > ν with dist( µ, ν ) = 1, and let β ∈ ∆ +re bethe (unique) positive real root such that ν = r β µ ; by Lemma 3.2.4, it suffices to showthat β = α or α . By Lemma 3.2.2, if µ = x m λ and m is even, then β = x l ( α ) or y l +1 ( α ) for some l ∈ Z even ≥ . Assume that β = x l ( α ) for some l ∈ Z even ≥ ; note that r β = ( r r ) l r ( r r ) l . We see from Lemma 3.2.4 that there exist a directed path µ = x m λ α ←− x m − λ α ←− · · · α ←− r β µ = ν of length 2 l + 1 from µ to ν in the Hasse diagram of W λ . Because dist( µ, ν ) = 1 byassumption, we obtain l = 0, and hence β = α . Assume that β = y l +1 ( α ) for some l ∈ Z even ≥ ; note that r ( r r ) l r ( r r ) l r . By the same reasoning as above, thereexists a direct path of length 2 l + 3 > µ to ν in the Hasse diagram of W λ .However, this contradicts the assumption that dist( µ, ν ) = 1. Similarly, we can showthat if µ = x m λ and m is odd, then β = α . Also, we can show the assertion for thecase that µ = y m λ in exactly the same way as above. This completes the proof of theproposition. Lemma 3.2.6.
For any rational number 0 < σ < µ, ν ∈ W λ such that µ > ν , there does not exist a σ -chain µ = µ > · · · > µ r = ν for ( µ, ν ) such that µ k = λ for some 0 ≤ k ≤ r . Proof.
Suppose that µ k = λ for some 0 ≤ k ≤ r . Note that r ≥ µ > ν . If k < r (resp., k > µ k +1 = r λ (resp., µ k − = r λ )since dist( µ k , µ k +1 ) = 1 (resp., dist( µ k − , µ k ) = 1) by the assumption of the σ -chain.Thus, we obtain σ = − σ h λ, α ∨ i ∈ Z (resp., σ = σ h λ, α ∨ i ∈ Z ), which contradictsthe assumption 0 < σ <
1. If k = 0 or k = r , it is clear that σ = − σ h λ, α ∨ i ∈ Z or σ = − σ h r λ, α ∨ i ∈ Z by Proposition 3.2.5. This also contradicts the assumption.Thus, the lemma has been proved.The next proposition follows immediately from Lemma 3.2.6 and the definition ofLS paths. 8 roposition 3.2.7. Let π = ( ν , . . . , ν s ; σ , . . . , σ s ) ∈ B ( λ ). If ν u = λ for some1 ≤ u ≤ s , then s = 1 and π = ( λ ; 0 , Proof of Theorem 3.2.1.
We show that every π ∈ B ( λ ) is connected to ( λ ; 0 , ∈ B ( λ )in the crystal graph of B ( λ ). Assume first that ι ( π ) = x m λ for some m ∈ Z ≥ . Weshow by induction on m that π is connected to ( λ ; 0 , m = 0, then the assertionfollows immediately from Proposition 3.2.7. Assume that m >
0. Define i := ( m is even) , m is odd);note that h x m λ, α ∨ i i < r i x m λ = x m − λ (see Lemma 3.2.4). By Lemma 2.2.6, ι ( e max i π ) = r i ι ( π ) = r i x m λ = x m − λ . By the induction hypothesis, e max i π is connectedto ( λ ; 0 , π .Assume next that ι ( π ) = y m λ for some m ∈ Z ≥ . Since κ ( π ) ≤ ι ( π ) by the definitionof an LS path, we see by Proposition 3.2.5 that κ ( π ) = y k λ for some k ≥ m . Henceit suffices to show that if π ∈ B ( λ ) satisfies that κ ( π ) = y k λ for some k ∈ Z ≥ , then π is connected to ( λ ; 0 , k = 0, then the assertion follows immediately fromProposition 3.2.7. Assume that k >
0. Define j := ( k is even) , k is odd);note that h y k λ, α ∨ j i > r j y k λ = y k − λ . By Lemma 2.2.6, κ ( f max j π ) = r j κ ( π ) = r j y k λ = y k − λ . By the induction hypothesis, f max j π is connected to ( λ ; 0 , π . Thus, we have proved Theorem 3.2.1. Throughout this section, we assume that a, b = 1 in (2.1). Recall that the sequences { p m } m ∈ Z ≥ and { q m } m ∈ Z ≥ are defined in (3.1) and (3.2), respectively. Lemma 4.1.1.
For each k ≥
0, the numbers p k and p k +1 are relatively prime. Also,the numbers q k and q k +1 are relatively prime. Proof.
We give a proof only for p k and p k +1 ; the proof for q k and q k +1 is similar. Supposethat the assertion is false, and let m be the minimum k ≥ p k and p k +1 havea common divisor greater than 1. Let d ∈ Z > be a common divisor of p m and p m +1 .Since ( p m +1 = bp m − p m − (if m is even) ,p m +1 = ap m − p m − (if m is odd) ,
9e can deduce that p m and p m − have the same common divisor d , which contradictsthe minimality of m . Thus, we have proved the lemma. Theorem 4.1.2. (1) Let 0 < σ < µ, ν ∈ W λ be suchthat µ > ν . If µ = µ > µ > · · · > µ t = ν is a σ -chain for ( µ, ν ), then t = 1.(2) An LS path π of shape λ = Λ − Λ is either of the form (i) or (ii):(i) ( x m + s − λ, . . . , x m +1 λ, x m λ ; σ , σ , . . . , σ s ), where m ≥ , s ≥
1, and 0 = σ < σ < · · · < σ s = 1 satisfy the condition that p m + s − u σ u ∈ Z for 1 ≤ u ≤ s − y m − s +1 λ, . . . , y m − λ, y m λ ; δ , δ , . . . , δ s ), where m ≥ s − , s ≥
1, and 0 = δ < δ < · · · < δ s = 1 satisfy the condition that q m − s + u +1 δ u ∈ Z for 1 ≤ u ≤ s − Proof. (1) Suppose that t ≥
2. Assume first that µ = x m λ ; by Lemma 3.2.6, we have m ≥
3. Since dist( µ , µ ) = dist( µ , µ ) = 1 by the definition of a σ -chain, we seeby Proposition 3.2.5 that µ = x m − λ and µ = x m − λ . Take i, j ∈ { , } such that µ = r i µ and µ = r j µ . Then, by Lemma 3.1.3, h µ , α ∨ i i = h x m λ, α ∨ i i = − p m , h µ , α ∨ j i = h x m − λ, α ∨ j i = − p m − . Since − p m and − p m − are relatively prime (see Lemma 4.1.1), there does not exist a0 < σ < − σp m and − σp m − are integers. Thiscontradicts our assumption that µ = µ > µ > · · · > µ t = ν is a σ -chain for ( µ, ν ).Similarly, we can get a contradiction also in the case of µ = y m λ for some m ∈ Z ≥ .Thus, we have proved (1).(2) Let π = ( ν , . . . , ν s ; σ , . . . , σ s ) ∈ B ( λ ). Assume first that ν s = x m λ for some m ≥
0. Since ν > ν > · · · > ν s = x m λ by the definition of an LS path, we see byProposition 3.2.5 that ν = x k λ, ν = x k λ, . . . , ν s − = x k s − λ for some k > k > · · · > k s − > m . Here we recall that there exists a σ s − -chain for( ν s − , ν s ) = ( x k s − λ, x m λ ) by the definition of an LS path. By (1), we see that the lengthof this σ s − -chain is equal to 1, which implies that dist( ν s − , ν s ) = dist( x k s − λ, x m λ ) = 1.Hence it follows from Proposition 3.2.5 that k s − = m + 1. Take i ∈ I such that x m λ = r i x m +1 λ . Then, by the definition of a σ s − -chain, we have σ s − h x m +1 λ, α ∨ i i ∈ Z .Since h x m +1 λ, α ∨ i i = − p m +1 by (3.3), we obtain p m +1 σ s − ∈ Z . By repeating thisargument, we deduce that k u = m + s − u and p m + s − u σ u ∈ Z for every 1 ≤ u ≤ s − π is of the form (i).Assume next that ν s = y m λ for some m ≥
0. Suppose that ( s ≥ ≤ u ≤ s − ν u +1 = y k λ for some k ≥
0, but ν u = x l λ for some l ≥
0. By thedefinition of an LS path, there exists a δ u -chain for ( ν u , ν u +1 ). Then, by (1), the lengthof this δ u -chain is equal to 1, which implies that dist( ν u , ν u +1 ) = dist( x l λ, y k λ ) = 1.By the Hasse diagram in Proposition 3.2.5, we see that ( l, k ) = (1 ,
0) or (0 ,
1) Since x λ = y λ = λ , it follows form Proposition 3.2.7 that s = 1, which contradicts s ≥ ν = y k λ, ν = y k λ, . . . , ν s = y k s λ = y m λ, ≤ k < k < · · · < k s − < k s = m . By the same argument as above, we deducethat k u = m − s + u and q m − s + u +1 δ u ∈ Z . Hence, π is of the form (ii). This completesthe proof of Theorem 4.1.2. As an application of Theorem 4.1.2, we give an explicit description of the root operators e i and f i , i = 1 , . First, let π ∈ B ( λ ) be of the form (i) in Theorem 4.1.2(2). We set C (1) u := m + s − X k = m + s − u ( σ m + s − k − σ m + s − k − )( − k p k + ξ k ,C (2) u := m + s − X k = m + s − u ( σ m + s − k − σ m + s − k − )( − k +1 p k + ξ k +1 , where { p m } m ∈ Z ≥ is define as (3.1), and ξ k := ( k is even , k is odd , for k ∈ Z ≥ . Then, wt( π ) = C (1) s Λ + C (2) s Λ . Note that ±h x u λ, α ∨ i i > ∓h x u λ, α ∨ i i > u ∈ Z ≥ . Thus we see(cf. (2.5)) that m πi = min { C ( i ) u | ≤ u ≤ s } . Let us give an explicit description of f i π . We set u := max { ≤ u ≤ s | C ( i ) u = m πi } ;if u = s , then f i π = . Assume that 0 ≤ u ≤ s −
1; we see that σ u is equal to t in(2.9). By fact (2.10), we deduce that t in (2.9) is equal to σ ′ u := σ u + 1 p m + s − u − ξ m + s − u − if i = 1, σ u + 1 p m + s − u − ξ m + s − u if i = 2,which satisfies σ u < σ ′ u ≤ σ u +1 ; notice that if σ ′ u = σ u +1 , then u = s −
1, and hence11 ′ u = σ s = 1 . We have f i π = ( x m + s λ, x m + s − λ, . . . , x m λ ; σ , σ ′ , σ , . . . , σ s ) if u = 0 and σ ′ u < σ u +1 , ( r i x m λ ; 0 ,
1) if u = 0 and σ ′ u = σ u +1 , ( x m + s − λ, . . . , x m λ ; σ , . . . , σ u − , σ ′ u , σ u +1 , . . . , σ s ) if u ≥ σ ′ u < σ u +1 , ( x m + s − λ, . . . , x m − λ ; σ , . . . , σ s − , σ s ) if u ≥ σ ′ u = σ u +1 . (4.1)Similary, we give an explicit description of e i π as follows. We set u := min { ≤ u ≤ s | C ( i ) u = m πi } ;if u = 0, then e i π = . Assume that 1 ≤ u ≤ s ; we see that σ u is equal to t in (2.7).By fact (2.8), we deduce that t in (2.7) is equal to σ ′ u := σ u − p m + s − u + ξ m + s − u if i = 1, σ u − p m + s − u + ξ m + s − u if i = 2,which satisfies σ u − ≤ σ ′ u ≤ σ u ; notice that if σ ′ u = σ u − , then u = 1 and hence σ ′ u = σ = 0. We have e i π = ( x m + s − λ, . . . , x m λ, x m − λ ; σ , . . . , σ s − , σ ′ s , σ s ) if u = s and σ u − < σ ′ u , ( r i x m λ ; 0 ,
1) if u = s and σ u − = σ ′ u , ( x m + s − λ, . . . , x m λ ; σ , . . . , σ u − , σ ′ u , σ u +1 , . . . , σ s ) if u ≤ s − σ u − < σ ′ u , ( x m + s − λ, . . . , x m λ ; σ , σ , . . . , σ s ) if u ≤ s − σ u − = σ ′ u . (4.2)Note that x − λ = y λ. Example 4.2.1.
Let π = ( r r r r λ, r r r λ, r r λ ; 0 , p , p , ∈ B ( λ );note that m = 2 and s = 3 in Theorem 4.1.2 (i). Let us compute f i π, i = 1 , , usingformula (4.1). If i = 1, then we have u = 2 , σ ′ u = p if a = 2 and u = 0 , σ ′ u = p if a ≥
3. (Note that p = 1 if b = 3.) Thus, f π = ( r r r r λ, r r r λ ; 0 , p ,
1) if a = 2 , b = 3,( r r r r λ, r r r λ, r r λ ; 0 , p , p ,
1) if a = 2 , b > r r r r r λ, r r r r λ, r r r λ, r r λ ; 0 , p , p , p ,
1) if a ≥ a = 2, then b >
3. If i = 2, then we have u = 1 , σ ′ u = p . Thus, f π = ( r r r r λ, r r r λ, r r λ ; 0 , p , p , . Next, let π ∈ B ( λ ) be of the form (ii) in Theorem 4.1.2 (2). By a similar argumentto above, we have the following explicit descriptions of f i π and e i π . We set D (1) v := m − s + v X k = m − s +1 ( δ k − m + s − δ k − m + s − )( − k q k + ξ k +1 ,D (2) v := m − s + v X k = m − s +1 ( δ k − m + s − δ k − m + s − ( − k +1 q k + ξ k , where { q m } m ∈ Z ≥ is define as (3.2). Thenwt( π ) = D (1) s Λ + D (2) s Λ . We have m πi = min { D ( i ) v | ≤ v ≤ s } . Let us give an explicit description of f i π . We set v := max { ≤ v ≤ s | D ( i ) v = m πi } ;if v = s , then f i π = . Assume that 0 ≤ v ≤ s −
1. we set δ ′ v := δ v + 1 p m − s + v +1+ ξ m − s + v if i = 1, δ v + 1 p m − s + v +1+ ξ m − s + v if i = 2.We have f i π = ( y m − s λ, y m − s +1 λ, . . . , y m λ ; δ , δ ′ , δ , . . . , δ s ) if v = 0 and δ ′ v < δ v +1 , ( r i y m λ ; 0 ,
1) if v = 0 and δ ′ v = δ v +1 , ( y m − s +1 λ, . . . , y m λ ; δ , . . . , δ v − , δ ′ v , δ v +1 , . . . , δ s ) if v ≥ δ ′ v < δ v +1 , ( y m − s +1 λ, . . . , y m − λ ; δ , . . . , δ s − , δ s ) if v ≥ δ ′ u = δ v +1 . (4.3)Note that y − λ = x λ. Similarly, we give an explicit description of e i π as follows. We set v := min { ≤ v ≤ s | D ( i ) v = m πi } ;13f v = 0, then e i π = . Assume that 1 ≤ v ≤ s . We set δ ′ v := δ v − q m − s + v + ξ m − s + v if i = 1, δ v − q m − s + v + ξ m − s + v if i = 2.We have e i π = ( y m − s +1 λ, . . . , y m λ, y m +1 λ ; δ , . . . , δ s − , δ ′ s , δ s ) if v = s and δ v − < δ ′ v , ( r i y m λ ; 0 ,
1) if v = s and δ v − = δ ′ v , ( y m − s +1 λ, . . . , y m λ ; δ , . . . , δ v − , δ ′ v , δ v +1 , . . . , δ s ) if v ≤ s − δ v − < δ ′ v , ( y m − s +2 λ, . . . , y m λ ; δ , δ , . . . , δ s ) if v ≤ s − δ v − = δ ′ v . (4.4) Acknowledgment.
The author is grateful to Professor Daisuke Sagaki, her supervisor, for suggestingthe topic treated in this paper and lending his expertise especially through the studyof the connectedness of LS paths as a crystal graph. Also, she thanks the referee forgiving her valuable comments.