Recollection of GKM theory

We first recall some facts in T-equivariant cohomology theory.

We follow the paper [Tym05]. Denote a n-dimensional torus by T , topologically T is homotopic to (S¹)ⁿ. Take ET to be a contractible space with a free T -action.

Define BT to be the quotient ET /T . The diagonal action of T on X × ET is free, since the action on ET is free. Define X ×_T ET to be the quotient (X × ET )/T . We define the equivariant cohomology of X to be

H_T^∗(X) = H^∗(X ×_T ET ).

When X is a point and T = G_m,

H_T^∗(X) = H^∗(pt ×_T ET ) = H^∗(ET /T ) = H^∗(BT ) = H^∗(CP^∞) ∼= k[t].

When T = (S¹)ⁿ,

H_T^∗(pt) = k[t₁, · · · , t_n] ∼= S(t^∗). (6.1)

So we can identify any class in H_T^∗(pt) as a function on the Lie algebra t of T . The map X −→ pt allows us to pull back each class in H_T^∗(pt) to H_T^∗(X), so H_T^∗(X) is a module over H_T^∗(pt).

Fix a projective variety X with an action of T . We say that X is equivariantly formal with respect to this T -action if E² = E^∞in the spectral sequence associated to the fibration X ×_T ET −→ BT .

When X is equivariantly formal with respect to T , the ordinary cohomology of X can be reconstructed from its equivariant cohomology. Fix an inclusion map i : X −→ X ×_T ET , we have the pull back map of cohomologies: H^∗(X ×_T ET ⁱ

∗

−→ H^∗(X). The kernel of i^∗ is Pn

s=1t_s· H_T^∗(X), where t_s is the generator of H_T^∗(pt)

(see (4)) and we view it as an element in H_T^∗(X) by pulling back the map X −→ pt.

Also i^∗ is surjective so H^∗(X) = H_T^∗(X)/ker(i^∗).

If in addition X has finitely many fixed points and finitely many one-dimensional orbits, Goresky, Kottwitz, and MacPherson show that the combinatorial data en-coded in the graph of fixed points and one-dimensional orbits of T in X implies a particular algebraic characterization of H_T^∗(X).

Theorem 11 (GKM, see [Tym05], [GKM97]). Let X be an algebraic variety with a T -action with respect to which X is equivariantly formal, and which has finitely many fixed points and finitely many one-dimensional orbits. Denote the one-dimensional orbits O₁, . . ., O_m. For each i, denote the the T -fixed points of O_i by N_i and S_i and denote the stabilizer of a point in O_i by T_i. Then the map H_T^∗(X)−→ H^l _T^∗(X^T) = ⊕_pi∈X^TH_T^∗(p_i) is injective and its image is

(

f = (fp1, . . . , fpm) ∈ M

fixed pts

S(t^∗) : fNi|ti = fSi|ti for each i = 1, . . . , m )

.

Here ti is the lie algebra of Ti.

6.5 Affine paving of Gr

_e^Π

(M ) when M is a representation of Q of type A

Definition 5 ([Tym07] 2.2). We say a space X is paved by affines if X has an order partition into disjoint X₁, X₂, · · · such that each finite unionSj

i=1X_i is closed in X and each Xi is an affine space.

A space with an affine paving has odd cohomology vanishing.

Proposition 7 ([Tym07], 2.3). Let X = S X_i be a paving by a finite number of affines with each X_i homeomorphic to C^di. The cohomology groups of X are given by H^2k(X) =L

{i∈I | d_i=k}Z.

The main observation is the following lemma.

Lemma 17. Let M be a representation of Q, where Q is of type A with all edges in E pointing to the right. Then the quiver Grassmannian Gr^Π_e(M ) is paved by affines for any dimension vector e.

We need a sublemma first.

Sublemma 1. Suppose X is paved by X_i’s. Let Y ⊂ X be a subspace. If for each i, Y_i = X_iT Y is ∅ or affine then Y = S Y_i is an affine paving.

Proof. S

i≤jY_i = S

i≤j(X_iT Y ) = (S_i≤jX_i)T Y is closed in Y since S_i≤jX_i is closed in X.

Before we give the proof of lemma17, let us introduce some notations about Schubert decomposition of Grassmannian, following [Shi85]. let V be an n-dimensional vector space over a field k. Let d be an integer such that 1 ≤ d < n. Let V^d be the direct sum of d copies of V and ∧^dV be the d-th alternating product of V . Let

π : V^d−→ ∧^dV

be the morphism defined by (v₁, · · · , v_d) → v₁∧ · · · ∧ v_d. Fix a basis {e₁, · · · , e_n} of V . Then we can identify V^d with the set of all d × n matrices over k by

(v₁, · · · , v_d) 7→ (x_i(j))1≤i≤d,1≤j≤n,

where v_i = Σ_1≤j≤nx_i(j)e_j, x_i(j) ∈ k. Let P(∧^dV ) be the projection. We have Grass(d, V ) = p(πV^d− {0}). Put

I = {α = (α₁, · · · , α_d) ∈ Z^d; 1 ≤ α₁ < · · · < α_d ≤ n}.

For α = (α₁, · · · , α_d) in I, put

D_α = {(x_i(j)) ∈ V^d; x_i(j) = 0forj < α_i(1 ≤ i ≤ d)},

C_α = {(x_i(j)) ∈ D_α; x_i(α_j) = δ_ij(1 ≤ i, j ≤ d)}.

Then S_α = pπC_αis the Schubert cell and p(πD_α−{0}) is the Schubert variety, which is the closure of S_α. We will consider the subvariety of Grassmannian Gr_d(kⁿ)^φthat is invariant under a nilpotent operator φ. Let λ be an ordered partition of n, i.e, an ordered sequence (λ₁, · · · , λ_r) of positive integers such that λ₁ + · · · + λ_r = n.

We represent λ by a Young diagram with rows consisting of λ₁, · · · , λ_r squares respectively.

Definition 6. Fix a Young diagram λ with n squares. Let d be an integer such that 1 ≤ d < n. A d-tableau is a Young diagram of type λ whose d squares are distinguished by .

Fill in the squares of λ with the numbers 1, 2, · · · , n in the order from the left column to the right and for each column from up to down. For example, if λ = (4, 3, 2), we have ^{7 4 2 1}

8 5 3 9 6

. Now we could identify α = (α1, · · · , αd) in I with the d-tableau of type λ whose α₁, · · · , α_d-th squares are . An nilpotent operator can be represented by a Young diagram using its Jordan normal form. We could choose our basis {e₁, · · · , e_n} to be a Jordan normal basis. Explicitly, for a Young diagram numbered above, we define φ corresponding to λ by φei = ej if λ contains

i j and φe_i = 0 if ⁱ lies on the most left column.

Definition 7. A d-tableau is said to be semistandard if every square on the left position to on the same row is .

Lemma 18. ([Shi85] lemma1.8, and see the proof there also) Let S_α^φbe the subspace invariant under φ, i.e

S_α^φ= {W ∈ S_α; φW ⊂ W },

where S_α is the Schubert cell corresponding to α. Then S_α^φ is nonempty if and only if α is semistandard.

Now we want to understand S_α^φ when α is semistandard. For that purpose, we introduce a notion called initial number.

Definition 8. For a semistandard d-tableau α = (α₁, · · · , α_i, · · · , α_d) of type λ, we say i is an initial number of α if the square on the right side of ^αi is not . Lemma 19. For a semistandard d-tableau α, put

C_α^φ= {v₁, · · · , v_d) ∈ D_α; φv_i = v_j if α contains ^αiαj, (1 ≤ i < j < d), when i is initial x_i(α_j) = δ_ij. Then the morphism pφ : D_α∼= Sα induces an isomorphism

C_α^φ∼= S_α^φ.

Proof. The lemma is given in [Shi85] lemma1.10 but I found there seems have a mistake. The original proof says: take an element p(w₁ ∧ · · · ∧ w_d) ∈ S_α^φ, where (w₁, · · · , w_d) ∈ C_α. Let (v₁, · · · , v_d) be an element in C_α^φ such that {(v₁, · · · , v_d} = {φ^hw_i : i⁰s are the initial numbers of α and h ≥ 0} − {0}. However, this element is not always in C_α. For example, when α is , (remember the number indexing basis is ^{4 2 1}

5 3 ) let v₂ = e₂+ e₃, then v₄ = e₄+ e₅, violating x₅(α₄) = 0. So we have to make a change to the statement to relax the condition that (v1, · · · , vd) ∈ Cα to D_α. After this change, the proof goes word by word.

Now we can give the proof of lemma17.

Proof of lemma 17. Let V = ⊕M_i be the underlying vector space and φ = ⊕_a∈Hφ_a be the nilpotent operator on V .

Let n = dimV and d = P e_i. From the above discussion, we know that Gr_d(kⁿ)^φ=F S_α^φ. We want to show S_α^φT Gr_e^Π(M ) is affine. Take x ∈ S_α^φT Gr_e^Π(M ), from lemma19, x = ∧(φ^hwi : i⁰s are the initial numbers of α and φ^hwi 6= 0).

We now show that for x = v₁ ∧ · · · ∧ v_d ∈ Gr^Π_e(M ), where v_i = e_i +P

j6=ix_i(j)e_j, if e_i ∈ M_t, we have v_i ∈ M_t and conversely, if for each i, there exists t such that v_i ∈ M_t, x ∈ Gr_e^Π(M ). We denote this t determined uniquely by i as t(i).

Since span(v₁, ..., v_d) is a direct sum of some N_i ⊂ M_i, we have P r_t(v_i) ∈ span(v₁, ..., v_d), where P r_t is the projection from V to M_t, and so P r_t(v_i) = P a_pv_p. Comparing the coefficient of ep, by the definition of Dα, we have ap = 0 for p 6= i. So we have P r_t(v_i) = v_i, which implies v_i ∈ M_t.

Note that {v₁, · · · , v_d} is determined by w_i where i is an initial number(and vice versa).

We have w_i ∈ M_t(i). Denote l(i) be the number on the left of i in the d-tableaus.

If i is the leftmost, set l(i) to be ∅, and set e∅ = 0. Write w_i = e_i+P x_ije_j, where ej ∈ Mti, we have φ^r(wi) = el^r(i) +P xijel^r(j). Since M is kQ-module, we have l^r(i) = l^r(j) hence v_ir ∈ M_t(ir) and x ∈ Gr_e^Π(M ). So we have S_α^φT Gr_e^Π(M ) is affine. Apply lemma 17, we are done.

6.6 Proof of lemma 15

Proof. For M given by C^d1 −→ C^{φ d2} and a choice of e = (e₁, e₂), denote X = Gr^Π_e(M ).

First, we define a torus action on X. Let I = kerφ. Choose a basis e₁, e₂, · · · , e_s¹ of I and extend it to a basis e₁, · · · , e_s, e_s+1, · · · , e_tof M₁. Let J be span{e_s+1, · · · , e_t} so the image of J is span {f_s+1, · · · , f_t}. We extend the basis {f_i = φ(e_i)} of the image of J to a basis (f_s+1, · · · , f_t, f_t+1, · · · , f_r) of M₂. Let K=span{f_t+1, ...f_r}.

we have M1 = I ⊕ J and M2 = φ(J ) ⊕ K.

Let I = {1, · · · , s}, J = {s + 1, · · · , t} and K = {t + 1, · · · , r}. Let tori T_I = G^I_m, T_J = G^J_m, T_K = G^L_m act on I, J ∼= φ(J ), K by compotentwise multiplication (For instance, T_I acts on I by (t₁, · · · , t_s)P a_ie_i =P a_it_ie_i and on J, K trivially).

Hence they act on M₁ = I ⊕ J and M₂ = φ(J ) ⊕ K. This induces an action of T = T_I × T_J × T_K on Gr^Π_e(M ). By lemma 3, Gr_e^Π(M ) is paved by affines so by proposition 3 it has odd cohomology vanishing therefore the spectral sequence associated to the fibration X ×_TET −→ BT converges at E²and X is equivariantly formal.

Denote by f the forgetful map H_T^∗(X)−→ H^{f ∗}(X). From §4.4 we have ker(f ) = PdimT

1 t_sH_T^∗(X). Since cⁱ(S₂⊕ Q₁) = f (cⁱ_T(S₂⊕ Q₁)), it suffices to prove cⁱ_T(S₂⊕ Q₁)) ∈ ker(f ) when i > e₂− e₁+ dimI.

To use GKM theorem, we need to know the one dimensional orbits and T -fixed points of X.

First, we see what X^T is. For a point p = (V₁, V₂) in X, in order to be fixed by T , V₁ and V₂ need to be spanned by some of basis vectors e_i and f_i. For a subset S of I (resp. J ), we denote by e_S (resp. f_S) the span {e_i|i ∈ S} (resp.

1I also use letter e for dimension vector, but it should be clear which I mean.

span{f_i|i ∈ S}). The T-fixed points in X consist of all V = (V₁, V₂), such that V₁ = e_AS B, V₂ = f_CS D, for some A ⊂ I, B ⊂ C ⊂ J and D ⊂ K.

For any point p = (V1, V2) in X^T, let V1 = eAS B, V2 = fCS D. Over p, Q1 = (I ⊕ J )/e_{AS B} is isomorphic to e(I\A)⊕(J \B) (The restriction of a T -equivariant bundle to a T-fixed point is just a T -module). So over X^T, we can decompose S₂⊕ Q₁ as follows:

the number of T-fixed points in X

l(H_T^∗(X)). (6.2)

By functorality of Chern class, l(cⁱ_T(S₂⊕ Q₁)) = cⁱ_T(S₂|_X^T ⊕ Q₁|_X^T)).

We compute the T -equivariant Chern class over X^T. For² each p,

c^p_T(S₂ ⊕ Q₁) = c^p_T(E₂⊕ E₁) = X for any i. The action of T on E₂ is actually the same on each T-fixed point. And at each point, cⁱ_T(E₂) is the i^th elementary symmetric polynomial of t_s, 1 ≤ s ≤ dimT . So by (5), it suffices to show that cⁱ_T(E₁) ∈ l(H_T^∗(X)).

2We always denote S and Q but indicate over which space we are considering.

Now we will see what 1-dimensional orbits are. Take an orbit O_i, in order to be 1 dimensional its closure must contain two fixed points. Let O_i = O_iS{N_i}S{S_i}, where Ni = (eAS B, fCS D) and Si = (eA⁰S B⁰, fC⁰S D⁰) are the fixed points. Oi is one dimensional whenever either AS B and A⁰S B⁰ differ by one element with CS D = C⁰S D⁰ or CS D and C⁰S D⁰ differ by one element with AS B = A⁰S B⁰. In the first case, we have some s ∈ AS B and s⁰ ∈ A⁰S B⁰, such that AS B \ s = A⁰S B⁰\ s⁰.

Notice that the annihilator for the lie algebra t_i in S(t^∗) is generated by t_s− t_s⁰, so by theorem 5, the condition along Oi for an element h ∈ H_T^∗(X^T) to be in im(l) is

(t_s− t_s⁰) | (h_Ni− h_Si).

But we have

cT(E1)|Ni−cT(E1)|Si = (1+ts0) Y

i∈IS C\(A S B)\{s⁰}

(1+ti)−(1+ts) Y

i∈IS C\(A⁰S B⁰)\{s}

(1+ti).

Note that IS C \ (A S B) \ {s⁰} = IS C \ (A⁰S B⁰) \ {s}, so t_s − t_s⁰ divides c_T(E₁)|_Ni− c_T(E₁)|_Si. We conclude that cⁱ_T(E₁) ∈ l(H_T^∗(X)).

The other case is similar.

C H A P T E R 7

PROOF OF ISOMORPHISM WHEN M IS A REPRESENTATION OF Q OF TYPE A

We first prove that Ψ is surjective.

Lemma 20. (a) Denote Y =Q Gr_ei(k^di) and X = Gr_e^Π(M ). Then Y \ X is paved by affines.

(b) Ψ is surjective.

Proof. (a). Let a be the number where the Young diagram of φY has a^th row as the first row from the bottom that does not have one block. For example, in the left diagram, a=4.

−

→

Define φ⁰ be the operator of V that corresponds to the diagram by moving the left most block A of the a^th row to the bottom in the diagram of φ_Y. Let M⁰ be the corresponding module and X⁰ be Gr_eM⁰.

We claim that X⁰ \ X is paved by affines.

By lemma 18, we have X =F

α∈IC_α, where I is the set of all semi-standard young

tableau in λ. Also we have Y⁰ =F

α∈I⁰C_α⁰, where I⁰ is the set of all semi-standard young tableau in λ⁰. If α contains block A, α is s.s in λ implies α⁰ is s.s in λ⁰. If α does not contain block A , α also does not contain any block in that row, so α is still s.s in λ⁰. So I ⊂ I⁰.

For α that contains block A, there are two types. Let E be the set of α that contains block A and some other block in the row of A. Let F be the set of α that contains block A but no other block in the row of A. Let G be the set of α that does not contain block A. So we have I = EF F F G = F F(E F G).

Take α ∈ F , in λ, the block A in α is not initial so the vector indexed by A is determined by the initial vector. In λ⁰, A is the last block so the vector indexed by A is the basis vector indexed by block A. In both case the vector indexed by A has been determined, so C_α = C_α⁰ when α ∈ F. step by step until X⁰ becomes Y, so we are done.

(b). By lemma 1, X is paved. With part(a) , we have the homology map from X to Y is injective hence Ψ is surjective since it is the dual of homology map in complex coefficient. (see 2.2 in[Tym07]).

We want to prove the two sides of Ψ have the same dimension as k-vector spaces and actually we will prove it for a more general setting.

Definition 9. For a Π-mod M Let V = ⊕M_i be the underlying vector space and φ = ⊕a∈Hφa be the nilpotent operator on V . We say M is I-compatible if there is a Jordan basis {v_ij} of V such that each v_ij is contained in some M_r.

Definition 10. For a Π-module M , if the Young diagram of the associated operator φ has one row, we call M one direction module.

Proposition 8. If M is I-compatible, it is a direct sum of one direction module with multiplicities.

Lemma 21. If M is I-compatible, the dimension of two sides of Ψ have the inequal-ity: dim k[a_ij, b_ik]/I(M, e)) ≤ χ(Gr^Π_e(M )), where χ is the Euler characteristic. We denote the ring on the left by R(M, e).

First we state a lemma due to Caldero and Chapoton.

Lemma 22 (see prop 3.6 in [CC04]). For Π-module M, N , we have

χ(Gr_g(M ⊕ N ) = X

d+e=g

χ(Gr_d(M ))χ(Gr_e(N )).

The following two lemmas are proved after the proof of lemma 19.

Lemma 23. R(M, e)/ < b_1(d1−e₁) >∼= R(M⁰, e).

Lemma 24. b_1(d1−e1)R(M, e) is a module over R(M⁰⁰, e −P

i∈Iα_i).

Proof of lemma 19. We index the basis vector according to the Young diagram of φ as before but slightly different: e_ij corresponds to the block of i^th row (from up to down) and j^th column (from right to left, which is the difference from before and this will cause the problem that two blocks in the same column but different row have different j but we will fix an i sooner so will not be of trouble). so φ(e_ij) = e_i(j−1).

The basis vector in M1 appears in the first column or in the last column. If it lies in the last, we can take the dual to make it in the first. So, there exist i such that e_i1 ∈ M₁.

Let λ⁰ be the young diagram removing the block of e_i1 from the original one and M⁰ be the corresponding module. Let λ⁰⁰ be the young diagram removing i^th row and M⁰⁰ be the corresponding module. Apply lemma 20, we have

χ(Gr^Π_e(M )) = χ(Gr_e(M⁰)) + χ(Gr_{e−P αi}(M⁰⁰)).

We count the dim of R(M, e) by dividing it into two parts.

dimR(M, e) = dimb_1(d1−e1)R(M, e) + dimR(M, e)/b_1(d1−e1)R(M, e). degree of s₁ goes down by by 1, meaning we have one more vanishing condition which is b_1(d1−e1)= 0.

Proof of lemma 22. In order to define a module structure on b_1(d1−e₁)R(M, e), we lift the element in R(M⁰⁰, e −P

i∈Iα_i) to R(M, e) (since the former is a quotient of

1The following proof is originally written for type A case in general, but later the author found for general type A, the proof of existence of affine paving does not work so here we only consider the special case where the module M is just a kQ-module. In this, the proof is much simpler.

the latter) and let it act on b_1(d1−e₁)R(M, e) by multiplication. We denote J to be the degree v(M, γ, e) part of <Q

i∈I+(t_i)^γiQ

i∈I−(s_i)^γi, γ ∈ Γ >.

We need to check it is independent of the choice of the lift:

b_1(d1−e1)J ⊂ I(M, e), where I(M, e) = J ⊕ I(M⁰⁰, e −X

By the dimension description, the image of Φ_γ−$1 is at least 1 dimensional bigger than the image of Φ_γ since φ_1m(e_i1) = e_im is in the image but for γ − $₁ (since φ_1mis not a summand of φ) the projection of img(φ) on V_m is zero, where m is the smallest number in Γ+.

Theorem 12. Ψ is an isomorphism when M is a kQ-module.

Proof. by lemma 18, Ψ is surjective so dim k[aij, bik]/I(M, e)) ≥ χ(Gr_e^Π(M )) and by lemma 19 this is an equality so the theorem follows.

C H A P T E R 8

A CONSEQUENCE OF THIS CONJECTURE

In section 3, we defined G(G, Y ) as moduli of maps of between pairs form (d, d^∗) to (G/Y, pt). This is actually a local version of (fiber at a closed point c) the global loop Grassmannian with a condition Y to a curve C, G^C(G, Y ). To a curve C, define G^C(G, Y ) over the ran space R_C with the fiber at E ∈ R_C:

G^C(G, Y )_E =^def map[(C, C − E), (G/Y, pt)].

Denote the map from G^C(G, Y ) to R_C remembering the singularities by π.

One can ask if π is (ind) flat for any G and (Y, pt). The case we are concerned is when G⁰ = G ×Q

wT_w and Y =Q

w(G/N^w)^{af f}. Let c ∈ C, λ_w, µ_w ∈ X∗(T )^W. In particular, we restrict G^C(G⁰, Y ) to C × c and denote the image under projection from G(G⁰) to G(G) by X. We have X is a closed subscheme of Gr_G,X×c. Explicitly, an R-point of GrG,X×c consists of the following data

• x : specR −→ C. Let Γ_xbe the graph of x. Let Γ_cbe the graph of the constant map taking value c.

• β a G-bundle on specR × C.

• A trivialization η : β₀ −→ β defined on specR × C − (Γ^η _xS Γ_c).

An R-point of X over C × c consists of an R-point of Gr_G,X×c subject to the condition: For every i ∈ I, the composition

η_i : β₀×^GV ($_i) −→ β ×^GV ($_i) −→ β ×^GV ($_i) ⊗ O(hγ, λ_wi · Γ_x+ hγ, µ_wi · Γ_c).

Corollary 3 (Given the conjecture). The T-fixed point subscheme of this family is flat.

Conjecture 3. T-fixed subschemes flatness imply flatness.

We take λw = −w0λ + wλ, then T S_λ^w_w = Y^λ. In this case the conjecture is proved to be true. This flatness is mentioned in [KMW16] remark 4.3 and will reduce the proof of reduceness of Y^λ to the case when λ is $_i for each i ∈ I.

1By lemma 20, and given the conjecture, Euler character is the same as total cohomology.

A P P E N D I X

PROOF OF LEMMA 16

For an expression s_im· · · s_i1 of an element w in W , we say it is j-admissible if hα_ia, s_ia−1· · · s_i1$_j ≥ 0 for any a ≤ m.

Lemma 25. For any element w ∈ W , any reduced expression of w is j-admissible.(since we will fix an j, we will omit j and just say admissible).

Proof. Since we are in the ADE case,

s_i$_i = −$_i+ X

h is adjacent to i

$_h. (A.1)

s_i$_h = $_h, for h 6= i. (A.2)

We use induction on the length of w. Suppose lemma holds when l(w) ≤ m. Take a reduced expression of w ∈ W with length m + 1 : w = s_im+1· · · s_i1. Suppose this expression is not admissible, we have hα_im+1, s_im· · · s_i1$_ji ≤ 0. Since hα_im+1, $_ji ≥ 0, and by (6),(7)

hαim+1, stγi ≥ hαim+1, γi.

unless t = i_m+1, there must exists k such that i_k= i_m+1.Let k be the biggest num-ber such that i_k= i_m+1.

In the case there is no element in the set {i_m, · · · , i_k+1} is adjacent to i_m+1 in the

Coxeter diagram, s_im+1commutes with s_im· · · s_ik+1. Therefore s_im+1s_im· · · s_ik+1s_im+1 = s_im+1s_im+1s_im· · · s_ik+1 = s_im· · · s_ik+1 so the w = s_im+1· · · s_i1 is not reduced, contra-diction.

In the case where for some t, i_tis adjacent to i_m+1, we will show we can reduce to the case we have only one such t. Suppose we have at least two elements i_t1, i_t2· · · , i_th such that they are all adjacent to i_m+1. Since hα_im+1, s_im+1· · · s_ik· · · s_i1$_ji ≤ 0 and h > 1, by (6), (7), we must have some i_u1, i_u2 such that they are adjacent to i_m+1. Since one point at most has 3 adjacent points we must have some i_ux = i_ty. Let p = iux = ity. Using spsim+1sp = sim+1spsim+1 we can move sim+1 in front of sty so the number h is reduced by 1. We could do this procedure until h=1. In this case we can rewrite the sequence before s_ik−1 using the braid relation between i_m+1 and i_t:

sim+1· · · sit· · · si_k = sim+1· · · sit· · · sim+1 = · · · sim+1sitsim+1· · · = · · · sitsim+1sit· · · . Set β = s_ik−1· · · s_i1$_j. By induction hypothesis, s_im+1s_its_im+1· · · and s_its_im+1s_it· · · are admissible. So hα_it, βi ≥ 0 and hα_im+1, βi ≥ 0. Again using (6) and (7) we have hsitsim+1β, αim+1i ≥ 0. Then hsim+1· · · si1$j, αim+1i = hsitsim+1β, αim+1i ≥ 0, contradicts with s_im+1· · · s_i1 is not admissible.

Proof of lemma 16. Set F₀ = {$_j}. Let F_m be the set which contains all w$_j, where l(w) ≤ m. We use induction. Suppose lemma 2 holds for γ ∈ F_m, we will prove lemma holds when γ ∈ F_m+1. For any γ = w$_j ∈ F_m+1, by lemma 1, w has a reduced admissible expression: w = s_im+1· · · s_i1. Denote i_m+1 by i and β by s_im· · · s_i1$_j. So γ = s_iβ, β ∈ F_m. Since s_im+1· · · s_i1 is admissible, hβ, α_ii ≥ 0.

Therefore hγ, αii = hsiβ, αii = −hβ, αii ≤ 0 and we can apply prop 4.1 in [BK12].

Then D_γ(M ) = D_si(siγ)(M ) = D_siγ(Σ_iM ) , where Σ_i is the reflection functor

defined in section 2.2 in [BK12].

Let A = {j | j is adjacent to i, j ∈ I} and M_A = ⊕_s∈AM_s. The i^th component of ΣiM is the kernel of the map ξ (Still see section 2.2 in [BK12] for the definition of ξ) from M_A to M_i. Since β ∈ F_m, by induction hypothesis, we can apply this

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