Tight Monomials and the Monomial Basis Property

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TIGHT MONOMIALS AND MONOMIAL BASIS PROPERTY

BANGMING DENG AND JIE DU

Abstract. We generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized Cartan matrices and a monomial basis property of those associated with symmetric (classical) Cartan matrices to their respective symmetrizable case. We then link the two by establishing that a tight monomial is necessarily a monomial deﬁned by a weakly distinguished word. As an application, we develop an algorithm to compute all tight monomials in the rank 2 Dynkin case.

The existence of Hall polynomials for Dynkin or cyclic quivers not only gives rise to a simple realization of the ±-part of the corresponding quantum enveloping algebras, but also results in interesting applications. For example, by specializing q to 0, degenerate quantum enveloping algebras have been investigated in the context of generic extensions ([20], [8]), while through a certain non-triviality property of Hall polynomials, the authors [4, 5] have established a monomial basis property for quantum enveloping algebras associated with Dynkin and cyclic quivers. The monomial basis property describes a systematic construction of many monomial/integral monomial bases some of which have already been studied in the context of elementary algebraic constructions of canonical bases; see, e.g., [15, 27, 21, 5] in the simply-laced Dynkin case and [3, 18], [9, Ch. 11] in general. This property in the cyclic quiver case has also been used in [10] to obtain an elementary construction of PBW-type bases, and hence, of canonical bases for quantum affine sln. In this paper, we will complete this program by proving this property for all finite types. The second purpose of the paper is to establish some relationship between the monomial basis property and tight monomials. Following Lusztig [17], a monomial which is also a canonical basis element is called a tight monomial. See also [19, 1] for further work on tight monomials. We first prove that the tight monomial criterion given in [22] for quantum enveloping algebras associated with symmetric generalized Cartan matrices works also for all symmetrizable ones. This criterion is built on Lusztig’s criterion for signed bases [16, Ch. 14]. Then we show in the finite type case that a tight monomial is necessarily a monomial associated with a weakly distinguished word. Thus, to test a monomial to be tight, it suffices to test monomials associated with weakly distinguished words. We further conjecture that monomials associated with distinguished words cover all tight monomials. We organize the paper as follows. Beginning with a quiver Q with automorphism σ, we first briefly review the relationship between representations of the path algebra A = kQ F over the algebraic closure k = Fq of Fq and the fixed point Fq-algebra A of the Frobenius

Date: November 17, 2008. 2000 Mathematics Subject Classification. 17B37, 16G20. Supported partially by the Australian Research Council, the NSF of China, and the Doctoral Program of Higher Education. The research was carried out while Deng was visiting the University of New South Wales. The hospitality and support of UNSW are gratefully acknowledged. 1 2 BANGMING DENG AND JIE DU morphism F = FQ,σ,q. The generalization of the criterion for tight monomials is presented in §2. From §3 onwards, we assume that Q is a Dynkin quiver. In §§3–5, we define the generic extension map ℘ for a Dynkin quiver Q with automorphism σ and use it to establish the monomial basis property for quantum enveloping algebras associated with (Q,σ). In the last three sections, we discuss the relationship between tight monomials and the monomial basis property. In §6, we prove that a tight monomial is necessarily a monomial associated with a weakly distinguished word, and in §7, we develop an algorithm to compute tight monomials of rank 2 and determine explicitly those of type B2. Finally, in the last section, we verify the conjecture for type B2 that tight monomials all arise from directed distinguished words and identify them as the canonical basis elements described in [30]. Throughout the paper, Fq denotes the finite field of q elements and k is the algebraic closure Fq of Fq. For an algebra B over a field, the category of finite dimensional left B-modules will be denoted by B-mod.

1. Preliminaries

Let Q = (Q0,Q1) be a ﬁnite quiver with vertex set Q0 and arrow set Q1, and let σ be an automorphism of Q, that is, σ is a permutation on the vertices of Q and on the arrows of Q such that σ(hρ)=hσ(ρ) and σ(tρ) = tσ(ρ) for any ρ ∈ Q1, where hρ and tρ denote the head and the tail of ρ, respectively. Recall from [6] that there is a Frobenius morphism FQ,σ;q on the path algebra A := kQ of Q over k = Fq deﬁned by

q F = FQ,σ;q : A −→A, xsps 7−→ xsσ(ps), s s X X where s xsps is a k-linear combination of paths ps. This gives an Fq-algebra P AF = {a ∈ A | F (a)= a}.

F If Q contains no oriented cycles, then A is a finite dimensional hereditary Fq-algebra. F Conversely, every finite dimensional hereditary basic Fq-algebra is isomorphic to A for some quiver Q with automorphism σ (see [6, Th. 6.5] or [7, Th. 9.3]). By [6, Prop. 4.2], there is a Frobenius twist functor ( )[1] : A-mod −→ A-mod, M 7−→ M [1] which is an equivalence of categories. Alternatively, let M be an A-module and FM : M → M be an arbitrary Frobenius map on the k-vector space M. We define the FM -twist of M to be the A-module M [FM ] such that M [FM ] = M as vector spaces and the A-module structure is given by −1 −1 a ∗ m := FM FQ,σ;q(a)FM (m) for all a ∈A,m ∈ M. By [7, Lem. 2.5], M [FM ] is isomorphic to M [1]. A module M ∈ A-mod is called F -stable if M =∼ M [1]. Moreover, for an F -stable A-module M, there is a Frobenius map FM on M satisfying

FM (am)= FQ,σ;q(a)FM (m) for all a ∈A,m ∈ M, TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 3 that is, M [FM ] = M as A-modules. Consequently, we obtain an AF -module

F FM M = M = {m ∈ M | FM (m)= m}. By [6, Th. 3.2], the correspondence M 7→ M F induces a bijection between the isoclasses of F -stable A-modules and those of AF -modules. Suppose that Q contains no oriented cycles. Let I be the set of isoclasses of simple mod- F ules in A -mod. (Note that I identifies with the set of σ-orbits in Q0.) The Grothendieck F F group K0(A ) of A -mod is then identified with the free abelian group ZI with basis I. F F Given a module M in A -mod, we denote by dim M the image of M in K0(A ), called F dimension vector of M. Then, for each i ∈ I, we have a simple A -module Si with dimension vector i. Hence, if dim M = i∈I xii, then xi is the number of composition factors isomorphic to Si in a composition series of M. The Euler from h−, −i : ZI × ZIP→ Z associated with (Q,σ) is defined by 1 hdim M, dim Ni = dim Fq HomAF (M, N) − dim Fq Ext AF (M, N), F for M, N ∈A -mod. For i ∈ I, let di = dim Fq EndAF (Si)= hi, ii, and for i, j ∈ I, define 2, if i = j, ci,j = 1 (hdim S , dim S i + hdim S , dim S i), if i 6= j. ( di i j j i

The matrix CQ,σ = (ci,j)i,j∈I is a symmetrizable generalized Cartan matrix with sym- metrization D = diag{di : i ∈ I}. Let g = g(CQ,σ) be the Kac–Moody algebra associated with CQ,σ and let U = Uv(g) be the corresponding quantized enveloping algebra over the fraction field Q(v) in indeterminate v. We are interested in the positive part U+ of U, which is by definition the Q(v)-subalgebra −1 of U generated by Ei, i ∈ I. Let Z = Z[v, v ] be the Laurent polynomial ring over Z. + Although the definition of U depends on a realization of CQ,σ, U depends only on CQ,σ + and is called the quantum algebra associated with CQ,σ (or (Q,σ)) in the sequel. Let U m + (m) Ei be the Z-subalgebra of U generated by divided powers Ei := ! for all i ∈ I, m ∈ N, [m]i a −a ! vi −vi di where [m]i = [1]i[2]i · · · [m]i with [a]i = −1 (vi = v ). vi−vi Let Ω be the set of all words in the alphabet I. For each word w = i1i2 · · · im ∈ Ω, t t i = (i1,... ,it) ∈ I , and a = (a1,... ,at) ∈ N , define monomials + Ew := Ei1 Ei2 · · · Eim ∈ U (1.0.1) (a) (a1) (a2) (at) + Ei := Ei1 Ei2 · · · Eit ∈ U . + + (a) t t Then U is spanned by {Ew}w∈Ω, while U is spanned by {Ei | i ∈ I , a ∈ N ,t ∈ N}. Thus, it would be interesting to ask how to extract monomial bases from these spanning sets. For quantum algebras associated with a Dynkin or cyclic quiver, the answer is given in [4, 5]. We will generalize this result in §§3-5 to include the nonsimply-laced Dynkin cases.

2. A Criterion for Tight monomials: the general case In [22], Reineke gave a criterion for a monomial to be tight (i.e., to be a canonical basis element) in a quantum enveloping algebra associated with a symmetric generalized Cartan 4 BANGMING DENG AND JIE DU matrix. We will see in this section that this criterion can be easily extended to all quantum enveloping algebras. As in §1, let Q be a quiver with automorphism σ. Suppose that Q contains no oriented cycles. Then CQ,σ is a symmetrizable generalized Cartan matrix. Moreover, the Euler form h−, −i associated with (Q,σ) gives a Cartan datum (I, ·) in the sense of [16, 1.1.1], where the bilinear form ZI × ZI → Z, (x, y) 7→ x · y is deﬁned by x · y = hx, yi + hy, xi, for x, y ∈ ZI.

Thus, for each i ∈ I, i · i = 2hi, ii = 2di. We will identify CQ,σ with its associated Cartan datum (I, ·) and speak of the quantum algebra U+ associated with (I, ·). + + + The algebra U admits an NI-grading U = ⊕x∈NI Ux such that Ux is spanned by all + monomials Ei1 · · · Eis with i1 + · · · + is = x. Given a homogeneous element x ∈ Ux , we write |x| = x. For each s > 2, there is a twisted product on the s-fold tensor product U+ ⊗···⊗ U+ given by s

|xi|·|yj| | {z }(x1 ⊗···⊗ xs)(y1 ⊗···⊗ ys)= v i>j x1y1 ⊗···⊗ xsys, P + where ⊗ = ⊗Q(v), and x1,... ,xs,y1,... ,ys are homogeneous elements in U . This algebra is called the s-fold graded tensor product of U+ and will be denoted by (U+)⊗s := U+⊗··· ⊗U+. Following [16, 1.2.2], there is a uniquee algebra homomorphism r : U+ → U+⊗U+ such e e that r(Ei)= Ei ⊗ 1 + 1 ⊗ Ei for each i ∈ I. Moreover, (r ⊗ 1)r = (1 ⊗ r)r : U+ −→ U+⊗U+⊗U+. e In general, for each s > 2, there is an algebra homomorphism e e (s) + + ⊗s ⊗s1 ⊗s2 r : U −→ (U ) , Ei 7−→ 1 ⊗ Ei ⊗ 1 . s +s =s e 1 X2 In particular, r = r(2). t t (a) For i = (i1,... ,it) ∈ I and a = (a1,... ,at) ∈ N , let Ei be the monomial deﬁned in (1.0.1). Using [16, 1.4.2] and an inductive argument (see [22, Lem. 2.5]), we have for each s > 2,

(s) (a) ηi,a (a1) (as) (2.0.2) r (Ei )= v Ei ⊗···⊗ Ei , a +···+a =a 1 X s t where ar = (ar1,... ,art) ∈ N , for 1 6 r 6 s, and

ηi,a = him, imiapmarm + (il · im)apmarl. 16m6t 16p

(3) (x0x00,y) = (x0 ⊗ x00, r(y)) for all x0,x00,y ∈ U+.

i·i/2 di 0 00 0 00 0 0 00 00 0 00 0 00 + Here vi = v = v and (x ⊗ x ,y ⊗ y ) := (x ,y )(x ,y ) for all x ,x ,y ,y ∈ U . Moreover, for each i ∈ I and a > 0, (2.1.1) a 1 va(a+1)/2 (E(a), E(a))= = i ∈ 1+ v−1Z[[v−1]] ∩ Q(v). i i −2m −1 a ! m=1 (1 − vi ) (vi − vi ) [a]i Y Observe that the sum in (2.0.2) is taken over the decompositions of a = a1 +· · ·+as each of which deﬁnes an s×t matrix A = (arm) with rows a1,... , as satisfying co(A)= a, where s s t t co(A) = ( p=1 ap1,... , p=1 apt). We also put ro(A) = ( m=1 a1m,... , m=1 asm).

t t DefinitionP 2.2. For anyP fixed i = (i1,... ,it) ∈ I and Pa = (a1,... ,atP) ∈ N , let Mi,a be the set of t × t matrices A = (arm) with entries in N satisfying the conditions ro(A) = co(A)= a and arm = 0 unless ir = im. Define a quadratic form q : Mi,a → Z by setting

q(A)= him, imiapmarm + (il · im)apmarl + hir, iriarmarl, 16m6t 16p

a1 − x 0 x M = {A | 0 6 x 6 min{a , a }}, where A = 0 a 0 , i,a x 1 3 x  2  x 0 a3 − x   and

q(Ax)= hi1, i1ia11a31 + hi3, i3ia31a33 + (i1 · i2)a22a31 + (i2 · i3)a13a22 + (i1 · i3)a13a31

+ hi1, i1ia11a13 + hi3, i3ia13a33 2 = 2hi1, i1ix(a1 − x) + 2hi3, i3ix(a3 − x)+ (i1 · i2) + (i2 · i3) xa2 + (i1 · i3)x .

4 (2) If i ∈ {(2, 1, 2, 1), (1, 2, 1, 2)} and a = (a1, a2, a3, a4) ∈ N , then

Mi,a = {Ax,y | 0 6 x 6 min{a1, a3}, 0 6 y 6 min{a2, a4}}, where

a1 − x 0 x 0 0 a2 − y 0 y Ax,y =   , x 0 a3 − x 0  0 y 0 a4 − y     6 BANGMING DENG AND JIE DU and q(Ax,y) = 2 hi1, i1ix(a1 − x)+ hi2, i2iy(a2 − y)+ hi3, i3ix(a3 − x)+ hi4, i4iy(a4 − y) 2 2 + (i1 · i2) + (i2 · i3) x(a2 − y) + (i1 · i3)x + (i1 · i4) + (i2 · i3) xy + (i2 ·i4)y + (i2 · i3) + (i3 · i4)y(a3 − x). (a) (a) (ar ) (ar ) Lemma 2.1 together with (2.0.2) allows us to compute (Ei , Ei ) in terms of (Eir , Eir ) (arm) (arm) M and (Eir , Eir ) for all A = (arm) ∈ i,a. The proof of the following result is similar to that of [22, Th. 2.2] for the symmetric case. However, we provide a proof for completeness. t Corollary 2.4. Keep the notation above. For i = (i1,... ,it) ∈ I and a = (a1,... ,at) ∈ Nt, we have

(a) (a) q(A) (arm) (arm) (Ei , Ei )= v (Eir , Eir ) M 6 6 A=(armX)∈ i,a 1 Yr,m t t (ar ) (ar ) q(A) (arm) (arm) = (Eir , Eir )+ v (Eir , Eir ), r=1 M 6 6 Y A=(arm)X∈ i,a\{Da} 1 Yr,m t where Da := diag(a1,... ,at). Proof. By Lemma 2.1(3) and (2.0.2), (a) (a) (a1) (at) r(t) (a) (Ei , Ei )= Ei1 ⊗···⊗ Eit , (Ei )

ηi,a (a1) (at) (a1) (at) = v Ei1 ⊗···⊗ Eit , Ei ⊗···⊗ Ei a1+···+at=a X ηi,a (ar ) (ar) = v Eir , Ei , a1+···+at=a 16r6t X Y a 6 6 (ar ) (ar) where r = (ar1,... ,art) for 1 r t. By Lemma 2.1(2), (Eir , Ei ) 6= 0 unless ar = ar1 + · · · + art and arm =0 if im 6= ir. This implies that the matrix A = (arm) with ! a [ar] a a M ( r) ir (ar ) rows 1,... , t is in i,a, and Ei = ! ! Eir . Therefore, [ar1]ir ···[art]ir

(a) (a) ηi,a (ar ) (ar) (Ei , Ei )= v Eir , Ei A=(a )∈M 16r6t rmX i,a Y ! [ar] = vηi,a ir E(ar ), E(ar ) ! ! ir ir [ar1] · · · [art] A=(a )∈M 16r6t ir ir rmX i,a Y ! ar(ar +1)/2 [ar] v = vηi,a ir ir [a ]! · · · [a ]! (v − v−1)ar [a ]! M 6 6 r1 ir rt ir ir ir r ir A=(armX)∈ i,a 1 Yr t arm(arm+1)/2 armarl v = vηi,a v 16l

+ (a) Let B be the canonical basis of U ; see Remark 4.5. Following [17], a monomial Ei is called tight if it belongs to B. We now can easily extend [22, Th. 3.2] to the general case. Theorem 2.5. Let U+ be the quantum algebra associated with a Cartan datum (I, ·). For t t (a) i = (i1,... ,it) ∈ I and a = (a1,... ,at) ∈ N , the monomial Ei is tight if and only if q(A) < 0 for all A ∈ Mi,a\{Da}. Proof. By [16, Ch. 14] (see also [22, Prop. 3.1]), (a) (a) (a) −1 −1 Ei ∈ B ⇐⇒ (Ei , Ei ) ∈ 1+ v Z[[v ]] ∩ Q(v). M Furthermore, by (2.1.1), for all A = (arm) ∈ i,a and 1 6 r, m 6 t, (arm) (arm) −1Z −1 Q (Eir , Eir ) ∈ 1+ v [[v ]] ∩ (v). The assertion then follows from Corollary 2.4. The following result is very useful in the determination of tight monomials; see §7 for the rank 2 case. t t (a) Corollary 2.6. Let i = (i1,... ,it) ∈ I and a = (a1,... ,at) ∈ N . Suppose Ei is tight. (ar ) (ar+1) (as) 6 6 6 Then the monomials Eir Eir+1 · · · Eis , for all 1 r s t, are also tight.

Proof. Write j = (ir,... ,is) and b = (ar,... ,as). Then

(b) (ar ) (ar+1) (as) Ej = Eir Eir+1 · · · Eis .

For each B ∈ Mj,b, define Da0 0 0 B = 0 B 0 ,   0 0 Da00 0 00e   where a = (a1,... ,ar−1) and a = (as+1,... ,at). It is easy to see that B ∈ Mi,a and (a) q(B) = q(B). Moreover, B = Da if and only if B = Db. Since Ei is tight, we get by Proposition 2.5 that e e e q(B)= q(B) < 0 for all B ∈ Mj,b\{Db}. Applying Proposition 2.5 again, we conclude that E(b) is tight, as desired. e j 3. The generic extension map associated with (Q,σ) One of the main ingredients in describing the monomial basis property is the generic extension map ℘ from the set Ω of all words in the index set I of simple representations to the set of isoclasses of all representations over Fq. Since representations of a Dynkin quiver Q over an arbitrary field are determined by their dimension vectors, we can simply define the map ℘ by sending a word w = i1i2 ...ir to the generic extension Si1 ∗ Si2 ∗···∗ Sir of simple representations Si1 ,Si2 ,... ,Sir of Q over k. This definition does not make sense if k is replaced by the finite field Fq. However, the theory developed in §1 can be used to generalize the definition of ℘. 8 BANGMING DENG AND JIE DU

From now on, we assume that Q is a (connected) Dynkin quiver, that is, the underlying graph of Q is a Dynkin graph (of type A, D or E). Suppose σ is an automorphism of Q. By a well-known result in [13, 12], the correspondence M 7→ dim M induces a bijection between the set of isoclasses of indecomposable AF -modules and the set of positive roots Φ+ = + + Φ (Q,σ) of the simple Lie algebra g = g(CQ,σ) associated with CQ,σ. For each α ∈ Φ , let F Mq(α) denote the corresponding indecomposable A -module. Thus, dim Mq(α) = α. By the Krull–Remak–Schmidt theorem, every AF -module M is isomorphic to

M(λ)= Mq(λ) := λ(α)Mq(α) + αM∈Φ for some function λ :Φ+ → N. Hence, the isoclasses of AF -modules are indexed by the set P = P(Q,σ) := {λ | Φ+ → N}, which is clearly independent of q. For convenience, we will view each α ∈ Φ+ as the function + Φ → N, β 7→ δαβ in P. It is shown in [20] that for any two A-modules M and N, there is a unique (up to isomorphism) extension G of M by N (i.e., 0 → N → G → M → 0) with minimal dimension of its endomorphism algebra EndA(G), which is denoted by G = M ∗ N and called generic extension of M by N. Moreover, for given A-modules L,M,N, (1) (L ∗ M) ∗ N =∼ L ∗ (M ∗ N), (2) L ∗ 0 =∼ L.

Thus, there is a monoid structure on the set MQ of isoclasses of A-modules with multiplication [M] ∗ [N] = [M ∗ N] and identity 1 = [0]. This monoid MQ has been studied in [20]. We have the following result (see [9, Porp. 11.1]). Lemma 3.1. Let M and N be F -stable A-modules. Then M ∗ N is also F -stable. Proof. Since the Frobenius twist functor ( )[1] : A-mod → A-mod is an equivalence of categories, we get (M ∗ N)[1] =∼ M [1] ∗ N [1]. From M [1] =∼ M and N [1] =∼ N it follows that (M ∗ N)[1] =∼ M ∗ N, that is, M ∗ N is F -stable.

In views of this lemma, the set MQ,σ of the isoclasses of F -stable A-modules becomes

a submonoid of M. For each vertex i ∈ I, deﬁne S{i} = Si ⊗Fq k. It is a semisimple A-module. Then, S{i}, i ∈ I, form a complete set of simple F -stable A-modules. The following proposition can be proved by using an argument similar to that in the proof of [20, Prop. 3.3]; see [9, Prop. 11.2].

Lemma 3.2. The monoid MQ,σ is generated by [S{i}], i ∈ I =Γ0. F Let MAF be the set of isoclasses of A -modules, i.e., MAF = {[Mq(λ)] | λ ∈ P}. F Applying the correspondence M 7→ M to MQ,σ yields F MAF = {[M ] | [M] ∈ MQ,σ}.

Thus, there is a map ℘ from Ω to P (or equivalently, to MAF ) deﬁned by (3.2.1) F [Mq(℘(w))] = [(S{i1}∗···∗ S{im}) ], for all w = i1i2 · · · im ∈ Ω. TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 9

This map ℘ :Ω → P, w 7→ ℘(w) is called the generic extension map associated with (Q,σ). −1 By Lemma 3.2, ℘ is surjective, and it induces a partition Ω = ∪λ∈P℘ (λ). Remark 3.3. Associated with the generic extension map ℘, one deﬁnes a generic extension graph as in [11, Def. 2.3]. It would be interesting to compare the generic extension graph with the corresponding crystal graph; see [11] for the type A case.

4. The monomial basis property Keep the notation introduced in the previous sections. Thus, (Q,σ) is a Dynkin quiver with automorphism, and A = kQ (resp., F = FQ,σ,;q) is the associated path algebra (resp., Frobenius morphism). Let U+ be the quantum algebra over Q(v) associated with (Q,σ) with Z-subalgebra U +. F M For A -modules M, N1,... ,Nm, let FN1,... ,Nm denote the number of ﬁltrations

M = M0 ⊇ M1 ⊇···⊇ Mm−1 ⊇ Mm = 0 such that Ms−1/Ms =∼ Ns for all 1 6 s 6 m. Ringel [25] shows that for λ, µ1,... ,µm ∈ P, λ there is an integral polynomial ϕµ1,...,µm (T ) ∈ Z[T ], called a Hall polynomial, such that for any ﬁnite ﬁeld Fq of q elements, ϕλ (q)= F Mq(λ) . µ1,...,µm Mq(µ1),...,Mq(µm)

By deﬁnition, the (twisted generic) Ringel–Hall algebra H = HZ (Q,σ) of (Q,σ) is the free Z-module with basis {uλ = u[M(λ)] | λ ∈ P}, and the multiplication is deﬁned by

hλ,µi π 2 uλuµ = v ϕλ,µ(v )uπ, πX∈P where hλ, µi = hdim M(λ), dim M(µ)i. For each i ∈ I, we write ui = u[Si]. Ringel [24, 26] proves that there is a Z-algebra isomorphism m + ∼ (m) (m) ui (4.0.1) U −→ H, Ei 7−→ ui := ! , (i ∈ I). [m]i In what follows, we simply identify U + with H under this isomorphism. In particular, + {uλ}λ∈P forms a Z-basis for U . λ λ For w = i1i2 · · · im ∈ Ω and λ ∈ P, let ϕw(T ) denote the Hall polynomial ϕi1,...,im (T ). λ Mq(λ) Thus, for each finite field Fq, ϕw(q) = F . Let further w be written in the tight Si1 ,... ,Sim form w = jm1 · · · jml , where j 6= j for all 1 6 i < l. We then denote ϕλ (T ) by 1 l i i+1 m1j1,...,mljl λ λ Mq(λ) γw(T ), i.e., γw(q) := F is the number of reduced filtrations m1Sj1 ,...,mlSjl

Mq(λ)= L0 ⊃ L1 ⊃···⊃ Ll−1 ⊃ Ll = 0 ∼ satisfying Li−1/Li = miSji for all 1 6 i 6 l. A word w is called weakly distinguished ℘(w) d if γw (T ) = T for some d ∈ N. We remark that distinguished words considered in ℘(w) [6, 7] are deﬁned by the condition γw (T ) = 1. The weak version considered here will become apparent in §6 as the words associated with tight monomials are necessarily weakly distinguished. 10 BANGMING DENG AND JIE DU

m1 ml To each word w = j1 · · · jl ∈ Ω in the alphabet I, where ji 6= ji+1 for all 1 6 i < l, we attach a monomial m(w) := E(m1)E(m2) · · · E(ml) ∈ U +. j1 j2 jl If we associate the word

(4.0.2) wi,a = i1 · · · i1 · · · it · · · it ∈ Ω

a1 at

t t (wi,a) (a) with i = (i1,... ,it) ∈ I and a = (a1|,...{z ,a} t) ∈| N{z,} then m = Ei in the notation of (1.0.1) whenever ij 6= ij+1 for all 1 6 j dim kHomA(X, M). Furthermore, the relation 6dg deﬁnes a partial order on the set MQ of isoclasses of A- modules, called the degeneration order (see [23, 2]).

The poset (MQ, 6dg) is compatible with the monoid structure on MQ (see [20]). More precisely, for given A-modules M,N,M 0, N 0, we have 0 0 0 0 (4.2.1) M 6dg M , N 6dg N =⇒ M ∗ N 6dg M ∗ N .

Restricting the order 6dg to the submonoid MQ,σ induces a partial order relation on MAF (and hence on P = P(Q,σ)). In other words, P has a partial order deﬁned by

λ 6 µ ⇐⇒ Mq(λ) ⊗ k 6dg Mq(µ) ⊗ k, λ,µ ∈ P. The next result follows directly from the deﬁnition and (4.2.1).

Proposition 4.3. For each w = i1i2 · · · im ∈ Ω, we have

ε1(w) λ 2 (4.3.1) Ew = v ϕw(v )uλ, λ6X℘(w) TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 11 where ε1(w)= 16r

m1 ml Now, for eachP word w = j1 · · · jl ∈ Ω in tight form, t −1 m(w) = E(m1) · · · E(ml) = [m ]! um1 · · ·uml . j1 jl r jr j1 jl r=1 Y t ! −ε2(w) t ! t Since r=1[mr]jr= v r=1[[mr]]jr , where ε2(w):= r=1 mr(mr − 1)djr /2, we have, by Proposition 4.3, that Q Q P t −1 (w) ! ε2(w) ε1(w)+ε2(w) λ 2 (4.3.2) m = [[mr[]jr v uw = v γw(v )uλ. r=1 Y λ6X℘(w) For λ ∈ P, let dim End(M(λ))−dim M(λ) (4.3.3) Eλ = v uλ If w is a directed distinguished word in ℘−1(λ) (see [5, §5] and [9, §11.2] for deﬁnition and existence and see §7 for certain examples), then [5, 6.6] and [9, Lem. 11.31] imply that

(4.3.4) ε1(w)+ ε2(w) = dim End(M(λ)) − dim M(λ). In particular, combining (4.3.2) and (4.3.3) yields (w) (4.3.5) m = E℘(w) + fµ,℘(w)Eµ, µ<℘X(w) where fµ,℘(w) ∈ Z. Thus, if we choose wλ to be directed distinguished for every λ ∈ P, then (4.3.5) gives

(wλ) (wµ) (4.3.6) Eλ = m + gµ,λm , µ<λX Now (4.3.5) and (4.3.6) imply

(4.3.7) ι(Eλ)= Eλ + rµ,λEµ. µ<λX where ι is the Z-algebra involution on U + deﬁned by + + (m) (m) −1 ι: U −→ U , Ei 7−→ Ei , v 7−→ v . −1 λ d Corollary 4.4. For a weakly distinguished word w ∈ ℘ (λ) with γw(T ) = T , (4.3.5) continues to hold. In particular, for any weakly distinguished word w ∈ ℘−1(λ),

ε1(w)+ ε2(w) + 2d = dim End(M(λ)) − dim M(λ). (w) s Proof. By (4.3.2), m = v E℘(w) + λ<℘(w) hµ,℘(w)Eλ, where

s = ε1(w)+ ε2(w) +P 2d − dim End(M(λ)) + dim M(λ). (w) −s 0 Applying ι and (4.3.7) yields m = v E℘(w) + λ<℘(w) hµ,℘(w)Eλ. Hence, s = 0, giving the ﬁrst assertion. The last assertion is a direct consequence of the ﬁrst one. P This proves part (2) of Theorem 4.1. 12 BANGMING DENG AND JIE DU

Remark 4.5. The above result shows that one may use a monomial basis associated with weakly distinguished words described in Theorem 4.1(2) to get the relation (4.3.7), and + hence, to construct the canonical basis B = {cλ | λ ∈ P} for U which is deﬁned uniquely by the conditions −1 −1 ι(cλ)= cλ, cλ ∈ Eλ + v Z[v ]Eµ. µ<λX In particular, a monomial m(w) lies in B (i.e., m(w) is a tight monomial) if and only if (w) −1 −1 m ∈ Eλ + µ<λ v Z[v ]Eµ. P 5. Proof of Theorem 4.1(1) By Proposition 4.3, it suﬃces to prove the following statement: λ −1 (∗) For any given λ ∈ P(Q,σ), we have ϕw(T ) 6= 0 for all w ∈ ℘ (λ). If σ = id, this result is a weaker version of [5, Prop. 6.2].

Proof. As before, let A = kQ be the path algebra of Q. It is well known that each A-module identiﬁes with a representation of Q over k. For a representation V = (Vi, Vρ) of Q over k, σ deﬁne V = (Wi,Wρ) by setting Wσ(a) = Va and Wσ(ρ) = Vρ for all a ∈ Q0 and all ρ ∈ Q1. In other words, as an A-module, σV is the module obtained by twisting the A-action on V via the algebra automorphism of A induced by σ. For each π ∈ P, set

Mq(π)k = Mq(π) ⊗Fq k,

Q0 which is an F -stable A-module. Let dim Mq(π)k = (ca) ∈ N . Up to isomorphism, we ca chρ×ctρ can identify Mq(π)k with a representation (Va, Vρ) of Q with Va = k and Vρ ∈ k for all a ∈ Q0 and ρ : tρ −→ hρ ∈ Q1. Then Mq(π)k = a∈Q0 Va, and there is a standard Frobenius map F on M (π) by taking (x ,... ,x ) ∈ V to (xq,... ,xq ) ∈ V . Since π q k 1 ca L a 1 ca a Mq(π)k is defined over Fq, we may suppose that all entries in matrices Vρ, ρ ∈ Q1, are chosen 0 in Fq. Then Mq(π) = {x ∈ Mq(π)k | Fπ(x) = x} becomes an FqQ-module. On the other [Fπ] hand, the Frobenius twist Mq(π)k = (Wa,Wρ) (relative to the Frobenius morphism FQ,σ;q [Fπ] on A) is given by Wσ(a) = Va and Wσ(ρ) = Vρ for a ∈ Q0 and ρ ∈ Q1. Hence, (Mq(π)k) σ [Fπ] σ identifies with (Mq(π)k). The F -stability of Mq(π)k implies that (Mq(π)k) = (Mq(π)k) σ is isomorphic to Mq(π)k. We fix an isomorphism ψπ : (Mq(π)k) → Mq(π)k, which is again 0 defined over Fq since both modules are so. Then Fπ := ψπFπ is also a Frobenius map [F 0 ] on Mq(π)k such that (Mq(π)k) π = Mq(π)k as A-modules. Then the fixed-point module 0 Fπ 0 F (Mq(π)k) = {x ∈ Mq(π)k | Fπ(x) = x} is an A -module, which can be identified with Mq(π). We first show the the following: ∼ λ Claim: For i ∈ I, λ, µ ∈ P with µ 6= 0, if Mq(λ)k = S{i} ∗ Mq(µ)k, then ϕi,µ(T ) 6= 0. From the definition, we have (see, for example, [28, Prop. 1])

Mq(λ) |ES ,M (µ)| F Mq(λ) = i q , Si,Mq(µ) |AutAF (Mq(µ))||AutAF (Si)| TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 13 where EMq(λ) denotes the set of pairs (f, g) of AF -module homomorphisms such that Si,Mq(µ) f g 0 −→ Mq(µ) −→ Mq(λ) −→ Si −→ 0 ∼ is an exact sequence. The fact Mq(λ)k = S{i} ∗ (Mq(µ)k) implies that

dim kMq(λ)k = dim kMq(µ)k + dim kS{i} and dim Fq Mq(λ)= dim Fq Mq(µ)+ dim Fq Si. F Since, up to isomorphism, Si is the unique A -module of dimension vector i, any injective homomorphism f : Mq(µ) → Mq(λ) induces an exact sequence f 0 −→ Mq(µ) −→ Mq(λ) −→ Si −→ 0. Thus, we obtain |X | F Mq(λ) = , Si,Mq(µ) |AutAF (Mq(µ))| F where X is the set of all injective A -module homomorphisms f : Mq(µ) → Mq(λ). Con- sequently, to show the claim, it suﬃces to prove that X is not empty for suﬃcient large q. 0 As indicated above, we have Frobenius maps Fλ, Fλ on Mq(λ)k and Frobenius maps 0 0 0 Fµ, Fµ on Mq(µ)k satisfying Fλ = ψλFλ and Fµ = ψµFµ. They induce a standard Frobenius map −1 Fλ,µ : HomA(Mq(µ)k, Mq(λ)k) −→ HomA(Mq(µ)k, Mq(λ)k), f 7−→ FλfFµ and a Frobenius map 0 0 0 −1 Fλ,µ : HomA(Mq(µ)k, Mq(λ)k) −→ HomA(Mq(µ)k, Mq(λ)k), f 7−→ FλfFµ . From the construction, via restriction of maps we can identify 0 Fλ,µ HomA(Mq(µ)k, Mq(λ)k) = HomAF (Mq(µ), Mq(λ)), whereas Fλ,µ 0 0 HomA(Mq(µ)k, Mq(λ)k) = HomFqQ(Mq(µ), Mq(λ)). 0 Further, we have Fλ,µ = ψ Fλ,µ, where −1 ψ : HomA(Mq(µ)k, Mq(λ)k) −→ HomA(Mq(µ)k, Mq(λ)k), f 7−→ ψλfψµ .

Let Y be the set of all injective A-module homomorphisms : Mq(µ)k → Mq(λ)k. It is an ∼ open subset of the aﬃne space V := HomA(Mq(µ)k, Mq(λ)k). Since Mq(λ)k = S{i}∗Mq (µ)k, Y Y 0 is not empty. Moreover, is stable under both Fλ,µ and Fλ,µ, and thus, 0 X Y Fλ,µ Y 0 = := {y ∈ | Fλ,µ(y)= y}.

For the standard Frobenius map Fλ,µ on V, we have by the theorem of Lang and Weil ([14]) that |Y Fλ,µ |≈ qt, where t = dim kV and ≈ means asymptotical behaviour. The standard Frobenius map Fλ,µ −1 on V induces a Frobenius map F on GLk(V) by taking g → Fλ,µgFλ,µ. By the known Lang Theorem, there is ψ1 ∈ GLk(V) such that −1 −1 −1 ψ = ψ1 F (ψ1 )= ψ1Fλ,µψ1 Fλ,µ. 14 BANGMING DENG AND JIE DU

0 −1 Y Y −1 Y Y Hence, Fλ,µ = ψFλ,µ = ψ1Fλ,µψ1 . Then 1 := ∩ ψ1 ( ) is open in V and ψ1( 1) = Y Y Y Y 0 ∩ψ1( ). It is easy to see that 1 and ψ1( 1) are stable under Fλ,µ and Fλ,µ, respectively, and that F 0 Y λ,µ Y Fλ,µ | 1 | = |ψ1( 1) |. F Y λ,µ Y Fλ,µ t By | 1 | ≈ | |≈ q , we ﬁnally get that 0 0 F F t |Y λ,µ | ≈ |ψ1(Y1) λ,µ |≈ q , X t λ that is, | |≈ q . We conclude that ϕi,µ(T ) 6= 0, proving the claim. ℘(w) Let w = i1i2 ...im ∈ Ω with m > 1. If m = 1, then clearly ϕw (T ) = 1 6= 0. ∼ Now suppose m > 2 and set w1 = i2 ...im, λ = ℘(w) and µ = ℘(w1). Then Mq(λ)k = λ Si1 ∗ Mq(µ)k, and hence, by the claim above, ϕi,µ(T ) 6= 0. On the other hand, by induction, µ we may suppose that ϕw1 (T ) 6= 0. Thus, there is some prime power q such that

F Mq(λ) 6=0 and F Mq(µ) 6= 0. Si1 ,Mq(µ) Si2 ,...,Sim This implies that F Mq(λ) = F Mq(λ) F Mq(π) 6= 0. Si1 ,Si2 ,...,Sim Si1 ,Mq(π) Si2 ,... ,Sim πX∈P ℘(w) Therefore, ϕw (T ) 6= 0.

6. Tight monomials and weakly distinguished words We are now ready to establish a relationship between tight monomials and weakly distinguished words. Assume again that Q is a Dynkin quiver with automorphism σ. Thus, CQ,σ is a classical Cartan matrix (of ﬁnite type). We keep all the notation in the previous sections. In particular, U+ is the quantum algebra associated with (Q,σ), which is identi- ﬁed with H ⊗Z Q(v) via Ei 7→ ui = ui ⊗ 1, ℘ is the generic extension map, and {cλ | λ ∈ P} is the canonical basis of U+.

t t (a) (b) Proposition 6.1. Let i, j ∈ I and a, b ∈ N . If both Ei and Ej are tight and ℘(wi,a)= (a) (b) ℘(wj,b), then Ei = Ej .

Proof. Let λ = ℘(wi,a)= ℘(wj,b). Applying (4.3.5) gives that

(a) −1 (b) 0 −1 Ei = φλ,µ(v, v )Eµ and Ej = φλ,µ(v, v )Eµ, 6 6 µXλ µXλ −1 0 −1 −1 −1 where φλ,µ(v, v ), φλ,µ(v, v ) ∈ Z[v, v ]. By the statement (∗) in §4, φλ,λ(v, v ) 6= 0and 0 −1 (a) (b) (a) (b) φλ,λ(v, v ) 6= 0. Since both Ei and Ej are tight, we must have Ei = cλ = Ej .

t Call a = (a1,... ,at) ∈ N is sincere if ai 6= 0 for all i.

t t (a) Theorem 6.2. Let i = (i1,... ,it) ∈ I and a = (a1,... ,at) ∈ N . If Ei is tight, then the word wi,a deﬁned in (4.0.2) is weakly distinguished. More precisely, we have ir 6= ir+1, TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 15

6 ℘(wi,a) d for all 1 r

(a) (a1) (at) (a1) (at) Ei = Ei1 · · · Eit = ui1 · · · uit t = var(ar −1) u · · · u ir [a1Si1 ] [atSit ] rY=1 t ar(ar −1) 16p

Paparhip, iri = hdimP M(λ), dim M(λ)i 6 6 1 Xp,r t = dim End(M(λ)) − dim Ext 1(M(λ), M(λ)), it follows that d0 = d. We now show that ir 6= ir+1 for all 1 6 r

(ar +ar+1)ir In other word, putting φ(T ) := ϕarir,ar+1ir (T ), µ P φ(T ) | ϕa1i1,... ,arir,ar+1ir+1 (T ) for all µ ∈ . 16 BANGMING DENG AND JIE DU

λ a 6 6 Hence, by (6.2.1), φ(T )|ϕa1i1,...,atit (T ). Consequently, φ(T )= T for some 0 a d. This is impossible since, for ar, ar+1 > 0 ar T ar+ar+1−s+1 − 1 ir hir,iri φ(T )= s , where Tir = T , Tir − 1 sY=1 is a non-constant Gaussian polynomial. λ Remark 6.3. It is very likely that a Hall polynomial of the form γw(T ) (obtained by counting the number of reduced ﬁltrations) cannot be a nonzero power of T . Thus, we conjecture that tight monomials are monomials associated with distinguished words. In fact, examples in §8 show that tight monomials are monomials associated with directed distinguished words (see [9, §11.2] for a deﬁnition).

7. Computing tight monomials: the rank 2 Dynkin case We now develop an algorithm for computing tight monomials in the Dynkin case of rank 2. As in Example 2.3, we consider in this section a quantum algebra U+ associated with a rank 2 Cartan datum (I, ·) of Dynkin type. Thus, I = {1, 2}. By (4.0.1), we can identify + U with the Ringel–Hall algebra H ⊗Z Q(v) via Ei 7→ ui = ui ⊗ 1. t t For i = (i1,... ,ii) ∈ I and a = (a1,... ,at) ∈ N , we are going to determine the tightness of the monomial (a) (a1) (at) U+ Ei = Ei1 · · · Eit ∈ . Without loss of generality, we suppose that all ai are positive, i.e., a is sincere. (a) First, it is direct to see from Remark 4.5 that Ei is tight whenever t 6 2. In other words, the monomials

(a1) (a1) (a1) (a2) (a1) (a2) E1 , E2 , E1 E2 , E2 E1 (a1 > 0, a2 > 0) are tight. Now we determine those for t > 3. The algorithm given below repeatedly uses the sets Mi,a and the quadratic forms described in Example 2.3.

7.1. Type A2. The canonical basis in this case is known and consists of tight monomials; see, e.g., [9, §11.7]. As a starting point of the algorithm, we give an independent construction. The Euler form in this case is defined by hx, yi = x1y1 + x2y2 − x2y1 for x, y ∈ ZI. Thus, the associated quiver Q has two vertices 1,2 and an arrow from 2 to 1. If t = 3, by Proposition 6.2, it suffices to consider i ∈ {(2, 1, 2), (1, 2, 1)}, and by Example 2.3(1), 2 q(Ax)= −2x + 2(a1 + a3 − a2)x, M 3 (a) for all Ax ∈ i,a with a = (a1, a2, a3) ∈ N . By Proposition 2.5, if Ei is tight, then q(Ax) < 0 for all 0 < x 6 min{a1, a3}. Setting x = 1 implies a1 + a3 6 a2. Hence, we obtain (a1) (a2) (a3) (a1) (a2) (a3) E2 E1 E2 (resp., E1 E2 E1 ) is tight ⇐⇒ a1 + a3 6 a2. (a) If t > 4, then all monomials Ei are not tight. This can be proved by an argument similar to the t > 5 case below. However, an application of the generic extension map defined in §3 shows that the founded tight monomials above already form the whole canonical basis. TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 17

Consider an arbitrary representation M of Q labelled by (a, b, c) in the sense that M =∼ aS1 ⊕bS12 ⊕cS2, where S12 is the indecomposable module with dimension vector (1, 1). If we b b+c a choose a = (b, b+c, a), then a1 +a3 6 a2 is equivalent to a 6 c and ℘(2 1 2 ) = (a, b, c). If c a+b b we choose a = (c, a+b, b), then a1+a3 6 a2 is equivalent to a > c and ℘(1 2 1 ) = (a, b, c). Moreover, a = c if and only if ℘(1c2a+b1b) = ℘(2b1b+c2a). Hence, we have proved the following (cf., [9, Prop. 11.35]). Proposition 7.1.1. The set

(a1) (a2) (a3) 3 (a1) (a2) (a3) 3 {E2 E1 E2 | a ∈ N , a1 + a3 6 a2} ∪ {E1 E2 E1 | a ∈ N , a1 + a3 6 a2} is a complete set of tight monomials of type A2 (and forms the canonical basis).

7.2. Type B2. The canonical basis in this case is constructed in [30]. However, not all tight monomials are identiﬁed in this construction; see Proposition 8.2. Consider the following quiver Q of type A3

b and let σ be the automorphism of Q deﬁned by exchanging two arrows. Then

F F 2 AF = (kQ)FQ,σ;q =∼ q q . 0 F 2 q F Up to isomorphism, there are two simple A -modules S1 and S2 with dim Fq S1 = 1 and dim Fq S2 = 2. Then the Euler form h−, −i : ZI × ZI → Z of (Q,σ) is given by

hx, yi = x1y1 + 2x2y2 − 2x2y1 for x, y ∈ ZI. Also, the associated Cartan datum (I, ·) is deﬁned by

x · y = hx, yi + hy, xi = 2x1y1 + 4x2y2 − 2x2y1 − 2x1y2, 2 −2 and the associated Cartan matrix CQ,σ = −1 2 has type B2. If t = 3, by Proposition 6.2, it suﬃces to consider i ∈ {(2, 1, 2), (1, 2, 1)}, and by Example 2.3(1), 2 −4x + 4(a1 + a3 − a2)x, if i = (2, 1, 2); q(Ax)= 2 (−2x + 2(a1 + a3 − 2a2)x, if i = (1, 2, 1), 3 for all Ax ∈ Mi,a with a = (a1, a2, a3) ∈ N , Since 0 6 x 6 min{a1, a3}, by Proposition 2.5,

(a1) (a2) (a3) E2 E1 E2 is tight ⇐⇒ a1 + a3 6 a2; (a1) (a2) (a3) ( E1 E2 E1 is tight ⇐⇒ a1 + a3 6 2a2. If t = 4, again by Proposition 6.2, we only need to consider i = (2, 1, 2, 1) or (1, 2, 1, 2). By Example 2.3(2), for each A = Ax,y ∈ Mi,a,

2 2 −(2x − y) − y + 4(a1 − a2 + a3)x + 2(a2 − 2a3 + a4)y, if i = (2, 1, 2, 1); q(A)= 2 2 (−x − (x − 2y) + 2(a1 − 2a2 + a3)x + 4(a2 − a3 + a4)y, if i = (1, 2, 1, 2). 18 BANGMING DENG AND JIE DU

Applying Proposition 2.5 gives that (7.0.1) (a1) (a2) (a3) (a4) E2 E1 E2 E1 is tight ⇐⇒ a1 + a3 6 a2 and a2 + a4 6 2a3; (a1) (a2) (a3) (a4) ( E1 E2 E1 E2 is tight ⇐⇒ a1 + a3 6 2a2 and a2 + a4 6 a3.

(a) If t > 5, then all monomials Ei are not tight. By Proposition 6.2 and Corollary 2.6, it suﬃces to show the case for i = (2, 1, 2, 1, 2) or (1, 2, 1, 2, 1). (a1) (a2) (a3) (a4) (a5) First, suppose that E2 E1 E2 E1 E2 is tight. Again, by Corollary 2.6, both (a1) (a2) (a3) (a4) (a2) (a3) (a4) (a5) E2 E1 E2 E1 and E1 E2 E1 E2 are tight. By (7.0.1), we obtain

a1 + a3 6 a2, a2 + a4 6 2a3, a3 + a5 6 a4.

Then the equalities a2 + a4 6 2a3 and a3 + a5 6 a4 imply a2 6 a3. This contradicts the (a1) (a2) (a3) (a4) (a5) inequality a1 + a3 6 a2. Hence, E2 E1 E2 E1 E2 is not tight. Second, suppose (a1) (a2) (a3) (a4) (a5) (a1) (a2) (a3) (a4) (a2) (a3) (a4) (a5) E1 E2 E1 E2 E1 is tight. Then both E1 E2 E1 E2 and E2 E1 E2 E1 are tight. Thus, by (7.0.1),

a1 + a3 6 2a2, a2 + a4 6 a3, a3 + a5 6 2a4. It follows that

2a2 + 2a4 > a1 + a3 + a3 + a5 > 2a2 + 2a4 + a1 + a5,

(a1) (a2) (a3) (a4) (a5) a contradiction. Hence, E1 E2 E1 E2 E1 is not tight. In a summary, we have the following result.

Proposition 7.2.1. The following is a complete list of tight monomials of type B2:

(a1) (a1) (a1) (a2) (a1) (a2) (1) 1, E1 , E2 , E1 E2 , E2 E1 , (a1) (a2) (a3) (a2) (a3) (a4) (2) E2 E1 E2 (a1 + a3 6 a2), E1 E2 E1 (a2 + a4 6 2a3), (a1) (a2) (a3) (a4) (3) E2 E1 E2 E1 (a1 + a3 6 a2 and a2 + a4 6 2a3), (a1) (a2) (a3) (a4) (4) E1 E2 E1 E2 (a1 + a3 6 2a2 and a2 + a4 6 a3), where a1, a2, a3, a4 ∈ N\{0}.

7.3. Type G2. This case can be computed by extending the algorithm above to the cases of t = 3, 4, 5, 6 and t > 7, and is much more complicated. For example, if i = (2, 1, 2, 1, 2, 1) 6 or i = (1, 2, 1, 2, 1, 2) and a = (ai) ∈ N , then each matrix A ∈ Mi,a has the form

x11 0 x12 0 x13 0 0 y 0 y 0 y  11 12 13 x 0 x 0 x 0 A = 21 22 23 ,  0 y 0 y 0 y   21 22 23 x31 0 x32 0 x33 0     0 y31 0 y32 0 y33     TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 19 where xij,yij ∈ N satisfy

3 3 3 3 xij = xji = a2i−1 and yij = yji = a2i for 1 6 i 6 3. Xj=1 Xj=1 Xj=1 Xj=1 We will treat this case separately elsewhere.

8. Identification of tight monomials of type B2

It is proved in [9, §11.7] that tight monomials of type A2 all arise from directed distinguished words. In this last section, we ﬁrst establish a similar result for the tight monomials of type B2 given in Proposition 7.2.1 and then identify them with the canonical basis elements described in [30]. Continue the type B2 case in the previous section and let

+ Φ (Q,σ)= {α1, 2α1 + α2, α1 + α2, α2} be the set of positive roots, where α1 and α2 are simple roots. The Auslander–Reiten quiver of AF = (kQ)FQ,σ;q in this case has the form

[P2] [S2]

[S1] [I1] where P2 and I1 are the projective cover and injective envelope of S2 and S1, respectively. Then the dimension vectors of P2 and I1 are 2α1 + α2 and α1 + α2, respectively. Following [9, §11.2], there are exactly ﬁve directed partitions Φ∗ of Φ+(Q,σ) given as follows: (1) (2) (1) Φ = {α1, 2α1 + α2}, Φ = {α1 + α2, α2}; (1) (2) (3) (2) Φ = {α1}, Φ = {2α1 + α2, α1 + α2}, Φ = {α2}; (1) (2) (3) (3) Φ = {α1, 2α1 + α2}, Φ = {α1 + α2}, Φ = {α2}; (1) (2) (3) (4) Φ = {α1}, Φ = {2α1 + α2}, Φ = {α1 + α2, α2}; (1) (2) (3) (4) (5) Φ = {α1}, Φ = {2α1 + α2}, Φ = {α1 + α2}, Φ = {α2}. Each λ ∈ P(Q,σ)= {µ | µ :Φ+(Q,σ) → N} can be also written as a 4-tuple (a, b, c, d) ∈ N4 with

a = λ(α1), b = λ(2α1 + α2), c = λ(α1 + α2), d = λ(α2). F In other words, the corresponding A -module Mq(λ) is given by

Mq(λ)= aS1 ⊕ bP2 ⊕ cI1 ⊕ dS1. For each λ = (a, b, c, d), the above five directed partitions define five directed distinguished words in the fibre ℘−1(λ):

b a+2b c+d c a b+c 2b+c d b a+2b c c d wλ,1 = 2 1 2 1 , wλ,2 = 1 2 1 2 , wλ,3 = 2 1 2 1 2 , a b 2b c+d c a b 2b c c d wλ,4 = 1 2 1 2 1 , wλ,5 = 1 2 1 2 1 2 . 20 BANGMING DENG AND JIE DU

Since the monomials corresponding to wλ,i for i = 3, 4, 5 are not tight for sincere λ, we only + consider the sets of monomials in U corresponding to wλ,1, wλ,2 for all λ. Let (b) (a+2b) (c+d) (c) M = {ma,b,c,d := E E E E | a, b, c, d ∈ N}, (8.0.2) 2 1 2 1 0 0 (a) (b+c) (2b+c) (d) M = {ma,b,c,d := E1 E2 E1 E2 | a, b, c, d ∈ N}. By Theorem 4.1, both M and M0 form bases for U +. Proposition 8.1. The tight monomials given in Proposition 7.2.1 form a subset of M∪M0. Proof. First, consider the tight monomials given in Proposition 7.2.1(3), (4). Since the inequalities a1 + a3 6 a2 and a2 + a4 6 2a3 imply a4 6 a3 and 2a1 6 a2, while a1 + a3 6 2a2 and a2 + a4 6 a3 imply a2 6 a3 6 2a2, it follows that (8.1.1) (a1) (a2) (a3) (a4) (a) ma,b,c,d = E2 E1 E2 E1 , if a = a2 − 2a1, b = a1, c = a4, d = a3 − a4; 0 (a1) (a2) (a3) (a4) ((b) ma,b,c,d = E1 E2 E1 E2 , if a = a1, b = a3 − a2, c = 2a2 − a3, d = a4. For those given in Proposition 7.2.1(2), we have by (8.1.1)(a)

(a1) (a2) (a3) (b) (a+2b) (d) E2 E1 E2 = ma,b,0,d = E2 E1 E2 ; (8.1.2) (a2) (a3) (a4) (a) (c+d) (c) ( E1 E2 E1 = ma,0,c,d = E1 E2 E1 . Finally, one checks easily that there exist a, b, c, d, e, f such that (b) (a+2b) ma,b,0,0(= E2 E1 ), if a2 > 2a1; (a1) (a2) (c+d) (c) E2 E1 = m0,0,c,d(= E2 E1 ), if a1 > a2;  0 (e+f) (2e+f) m0,e,f,0(= E2 E1 ), if a1 6 a2 6 2a1. The remaining cases are clear. We now identify the tight monomials given in Proposition 7.2.1 with the canonical basis elements computed by Xi. For nonnegative integers a, b, c, d, the monomials (a) (d) E1 E2 , (b) (a+2b) (c+d) (c) E2 E1 E2 E1 (c + d 6 a + b and a + 2b 6 c + 2d), (a) (b+c) (2b+c) (d) E1 E2 E1 E2 (a 6 c and d 6 b), are all the tight monomials appeared as the canonical basis elements described in [30, Th. 2.2]. Thus, the tight monomials in Proposition 7.2.1(1),(3),(4) appear in monomial form in Xi’s basis. However, most of the tight monomials in Proposition 7.2.1(2) do not appear in monomial form. We now use the representation-theoretic approach to identify them. We follow the notation introduced in (8.1.2). Thus, by Proposition 7.2.1(2), ma,0,c,d = (a) (c+d) (c) (b) (a+2b) (d) E1 E2 E1 (resp., ma,b,0,d = E2 E1 E2 ) is tight if a 6 c + 2d (resp., d 6 a + b). m m For m ∈ N and n ∈ Z, let be obtained from by replaced v by v2. n v2 n h i h i Proposition 8.2. Let a, b, c, d be nonnegative integers and keep the notation above. TIGHTMONOMIALSANDMONOMIALBASISPROPERTY 21

(1) If a 6 c and c > 1, then

l d + l − 1 (a) (c−l) (c) (d+l) ma,0,c,d = (−1) E1 E2 E1 E2 . l v2 6 6 0Xl c h i (2) If 1 6 c 6 a 6 c + d, then

m+l a − c + m − 1 d + m + l − 1 (a+m) (c−l) (c−m) (d+l) ma,0,c,d = (−1) E1 E2 E1 E2 . m l v2 6 6 0 Xm,l c h ih i (3) If d 6 b and b > 1, then

m l a + m − 1 (a+m) (b) (2b−m) (d) a,b,0,d = (−1) m E1 E2 E1 E2 . 6 6 0 Xm 2b h i (4) If 1 6 b 6 d 6 b + a/2, then

m+l a + 2l − m − 1 d − b + l − 1 (a+m) (b−l) (2b−m) (d+l) ma,b,0,d = (−1) E1 E2 E1 E2 . m l v2 06m62b 0X6l6b h ih i Proof. We only prove (1). Statements (2)–(4) can be proved similarly. Let λ = (a, 0, c, d) ∈ P(Q,σ), i.e., M(λ) = aS1 ⊕ cI1 ⊕ dS2. Since ma,0,c,d is tight, we have −1 ma,0,c,d ≡ Eλ mod v L , −1 + where L is the Z[v ]-submodule of U spanned by Eµ, µ ∈ P(Q,σ); see Remark 4.5. On the other hand, we denote the element in the right hand side of the equality in (1) by Xa,c,d. By [30, Th. 2.2], Xa,c,d is a canonical basis element and (a) 0 (c) (d) −1L Xa,c,d ≡ E1 (E12) E2 mod v , 0 (c) 0 c ! where (E12) = (E12) /[c] with 0 −2 E12 = E2E1 − v E1E2.

By the deﬁnition of the multiplication in HZ (Q,σ), we have 0 −2 −2 E12 = E2E1 − v E1E2 = u2u1 − v u1u2 −2 −2 −2 = v (u[I1] + u[S1⊕S2]) − v u[S1⊕S2] = v u[I1]. Thus, 0 (c) −2c ! −2c c2−c c2−3c (E12) = v u[I1]/[c] = v v u[cI1] = v u[cI1]. It then follows that (a) 0 (c) (d) a2−a+c2−3c+2d2−2d E1 (E12) E2 = v u[aS1]ucI1]u[dS2] a2−a+c2−3c+2d2−2d+ac+2cd = v uλ. Since dim End(M(λ)) = a2 + c2 + 2d2 + ac + 2cd and dim M(λ)= a + 3c + 2d. Thus, by (4.3.3), (a) 0 (c) (d) dim End(M(λ))−dimM(λ) E1 (E12) E2 = v uλ = Eλ 22 BANGMING DENG AND JIE DU and, hence, (a) 0 (c) (d) −1L ma,0,c,d ≡ Eλ ≡ E1 (E12) E2 ≡ Xa,c,d mod v . We ﬁnally conclude that ma,0,c,d = Xa,c,d, as required.

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School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. E-mail address: [email protected]

School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Aus- tralia. E-mail address: [email protected] Home page: http://web.maths.unsw.edu.au/∼jied