Hecke algebras of simply-laced type with independent parameters

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.


Introduction
Let W := S : (st) m st = 1, ∀s, t ∈ S be a Coxeter group generated by a finite set S with relations (st) m st = 1 for all s, t ∈ S, where m ss = 1 for all s ∈ S and m st = m ts ∈ {2, 3, . . .} ∪ {∞} for all distinct s, t ∈ S. Given a parameter q in a field F, the (Iwahori-)Hecke algebra H S (q) of the Coxeter system (W, S) is the (unital associative) algebra over F generated by {T s : s ∈ S} with • quadratic relations (T s − 1)(T s + q) = 0 for all s ∈ S, and • braid relations (T s T t T s · · · ) m st = (T t T s T t · · · ) m st for all s, t ∈ S.
Here (aba · · · ) m denotes an alternating product of m terms. The Hecke algebra H S (q) is a one-parameter deformation of the group algebra FW of W. It has an F-basis {T w : w ∈ W} indexed by W, where T w := T s 1 · · · T s ℓ if w = s 1 · · · s ℓ is a reduced (i.e., shortest) expression in the generators of W. The Hecke algebra H S (q) naturally arises in different ways and has significance in many areas (see, e.g., Lusztig [25]). Tits showed that, if W is finite, F = C is the field of complex numbers, and q ∈ C is neither zero nor a root of unity, then the Hecke algebra H S (q) is semisimple and isomorphic to the group algebra CW. The representation theory of Hecke algebras at roots of unity has been studied to some extent, with connections to other topics found (see Geck and Jacon [12]), but has not been completely determined yet even in type A (see Goodman and Wenzl [13]).
Another interesting specialization of H S (q) is the 0-Hecke algebra H S (0), which is different from but closely related to the group algebra of W. It was used by Stembridge [30] to give a short derivation for the Möbius function of the Bruhat order of the Coxeter group W and its parabolic quotients.
For a finite Coxeter system (W, S), Norton [26] studied the representation theory of H S (0) over an arbitrary field F using the triangularity of the product in H S (0). Her main result is a decomposition of H S (0) into a direct sum of 2 |S| many indecomposable submodules; this decomposition is similar to the decomposition of the group algebra of W (over the field of rational numbers) by Solomon [28]. Norton's result provided motivations to later work of Denton, Hivert, Schilling, and Thiéry [8] on the representation theory of finite J-trivial monoids, as H S (0) is a monoid algebra of the 0-Hecke monoid T w | q=0 : w ∈ W , an example of J-trivial monoids.
In type A, Krob and Thibon [22] discovered an important correspondence from representations of 0-Hecke algebras to quasisymmetric functions and noncommutative symmetric functions. This correspondence is analogous to the classical Frobenius correspondence from complex representations of symmetric groups to symmetric functions. Duchamp, Hivert, and Thibon [10] constructed the quiver of the 0-Hecke algebra H n (0) of type A n−1 and showed that H n (0) is of infinite representation type for n ≥ 4. Tewari and van Willigenburg [31,32] and König [21] studied connections between H n (0) and a new basis of quasisymmetric functions called the quasisymmetric Schur functions. After further investigation of combinatorial aspects of the representation theory of H n (0) [15,16] using the correspondence of Krob and Thibon, we recently extended this correspondence from type A to type B and type D [18,19].
Motivated by the similarities and differences between various specializations of the Hecke algebra H S (q), we generalized its definition from a single parameter q to multiple independent parameters and studied the resulting algebra in recent work [17]. The Hecke algebra H S (q) of a Coxeter system (W, S) with independent parameters q = (q s ∈ F : s ∈ S) ∈ F S is the F-algebra generated by {T s : s ∈ S} with • quadratic relations (T s − 1)(T s + q s ) = 0 for all s ∈ S, and • braid relations (T s T t T s · · · ) m st = (T t T s T t · · · ) m st for all s, t ∈ S.
We constructed a basis for H S (q) when (W, S) is simply laced and characterized when H S (q) is commutative. In type A, a commutative Hecke algebra H S (q) has its dimension given by a Fibonacci number and its representation theory has interesting features analogous to the representation theory of both symmetric groups and their 0-Hecke algebras.
In this paper we investigate the representation theory of the (not necessarily commutative) algebra H S (q) when (W, S) is a simply-laced Coxeter system. Our result shows an interesting combination of the representation theory of Coxeter groups and 0-Hecke algebras, but there are certain features of the representation theory of H S (q) that are unlike both symmetric groups and 0-Hecke algebras. For example, restrictions of projective H S (q)-modules are not projective in general.
We do not consider roots of unity here, but allowing these parameters would still give interesting algebras whose representation theory is yet to be determined. Although we focus on simply-laced Coxeter systems, some of the preliminary results in Section 2 are valid for all finite Coxeter systems, and it may be possible to extend our results to non-simply-laced Coxeter systems.
This paper is structured as follows. In Section 2 we review the representation theory of finite dimensional algebras, finite Coxeter groups, and 0-Hecke algebras, and also develop some basic results for later use. Next, in Section 3 we study the structure of the algebra H S (q) and give a formula for its dimension. In Section 4 we construct the projective indecomposable H S (q)-modules and simple H S (q)-modules, and determine the Cartan matrix and (ordinary) quiver of H S (q). In Section 5 we obtain formulas for induction and restriction of representations between H R (q) and H S (q| R ) for R ⊆ S, and verify a two-sided duality between induction and restriction. Lastly, we give some remarks and questions in Section 6.

Preliminaries
2.1. Representations of algebras. We first review some general results on representations of algebras; see references [3,7,11] for more details. All algebras and modules in this paper are finite dimensional.
Let A be an (unital associative) algebra over a field F. We say A is of finite [or infinite, resp.] representation type if the number of non-isomorphic indecomposable A-modules is finite [or infinite, resp.]. Let M be a There exists a direct sum decomposition A = P 1 ⊕ · · · ⊕ P k where P 1 , . . . , P k are indecomposable A-submodules. For i = 1, 2, . . . , k, the radical rad(P i ) is the unique maximal A-submodule of P i and C i := top(P i ) is simple [3,Proposition I.4.5 (c)]. Moreover, every projective indecomposable A-module is isomorphic to P i for some i and every simple A-module is isomorphic to C j for some j.
The algebra A is basic if P 1 , . . . , P k are pairwise non-isomorphic. In general, we may assume, without loss of generality, that {P 1 , . . . , P r } is a complete set of pairwise non-isomorphic projective A-modules and {C 1 , . . . , C r } is a complete set of pairwise non-isomorphic simple A-modules for some r ≤ k. The is the multiplicity of the simple module C i = top(P i ) among the composition factors of the projective indecomposable module P j .
The Grothendieck groups G 0 (A) and K 0 (A) of A are free abelian groups with bases {C 1 , . . . , C r } and {P 1 , . . . , P r }, respectively. If 0 → L → M → N → 0 is a short exact sequence of A-modules [or projective A-modules, resp.], then M is identified with L + N in G 0 (A) [or K 0 (A), resp.]. If A is a semisimple algebra then G 0 (A) = K 0 (A).
If M and N are two A-modules then define M, N := dim F Hom A (M, N). Since C i , C j = δ i,j for all i and j by Schur's Lemma and since f (rad(M)) ⊆ rad(N) for any f ∈ Hom A (M, N), we have where δ is the Kronecker delta. Taking M = P i gives the duality between G 0 (A) and K 0 (A). We next provide some basic results on representations of algebras for later use.
Proof. Part (i) and Part (ii) follow from a standard result [11,Theorem 2.26] and its proof. Applying (i) gives rad 2 Proposition 2.2. Let A = P 1 ⊕ · · · ⊕ P k and A ′ = P ′ 1 ⊕ · · · ⊕ P ′ ℓ be direct sum decompositions of two algebras A and A ′ into indecomposable submodules. Then    Under certain circumstances, e.g., when A and B are group algebras over the complex field C, one has . This is known as the Frobenius reciprocity. The other possible adjunction holds for the (complex) group algebras of the symmetric groups and their 0-Hecke algebras (over any field F), giving the duality between certain graded Hopf algebras (see Section 2.4). Next, recall that a quiver Q is a directed graph possibly with loops and multiple arrows between two vertices. Its path algebra CQ has a basis consisting of all paths in Q and has multiplication given by concatenation of paths. The arrow ideal R Q is the two-sided ideal of CQ generated by all arrows in Q. A representation of Q is a CQ-module. Gabriel's theorem classifies connected quivers of finite representation type as type A n , D n , E 6 , E 7 , and E 8 , meaning that these quivers do not contain oriented cycles and their underlying undirected graphs are given by Coxeter diagrams of the corresponding types (cf. Section 2.2).
Let A be a finite dimensional C-algebra whose projective indecomposable modules are P 1 , . . . , P r and let C i := top(P i ) for all i. The (ordinary) quiver Q A of A is the direct graph with vertices C 1 , . . . , C r such that the number of arrows from C i to C j is the multiplicity of C j among the composition factors of rad(P i )/ rad 2 (P i ). In particular, the quiver of a semisimple algebra A consists of isolated vertices.
If A is a basic algebra then there exists an ideal I of the path algebra CQ A such that A ∼ = CQ A /I and I ⊆ R 2 , where R is the arrow ideal of Q A [3,Theorem II.3.7]. If A is not basic then there is a basic algebra A b such that the categories of finitely generated modules over A and A b are equivalent [3, Corollary I. 6.10] and the quiver of A is the same as the quiver of A b (cf. Li and Chen [23, Proposition 1.2]).
Assume A 1 and A 2 are two algebras whose quivers Q 1 and Q 2 are loopless. The quiver of A 1 ⊗ A 2 is the tensor product Q 1 ⊗ Q 2 of Q 1 and Q 2 , a loopless quiver defined below: its vertex set is the Cartesian product of the vertex sets of Q 1 and Q 2 , and the number of arrows from (u 1 , the number of arrows from u 1 to v 1 , if u 1 = v 1 and u 2 = v 2 , the number of arrows from u 2 to v 2 , if u 1 = v 1 and u 2 = v 2 , zero, otherwise.

Coxeter groups and their representation theory.
We recall some basic definitions and results on Coxeter groups from Björner and Brenti [5]. A Coxeter group is a group W generated by a finite set S with quadratic relations s 2 = 1 for all s ∈ S and braid relations (sts · · · ) m st = (tst · · · ) m st for all distinct s, t ∈ S, where m st = m ts ∈ {2, 3, . . .} ∪ {∞} and (aba · · · ) m denotes an alternating product of m terms. Let (W, S) be a Coxeter system and let w ∈ W. We say that w = s 1 · · · s k is a reduced expression of w if s 1 , · · · , s k ∈ S and k is as small as possible; the smallest k is the length ℓ(w) of w. The descent set of w is defined as D(w) := {w ∈ S : ℓ(ws) < ℓ(w)} and its elements are called the descents of w. One has s ∈ D(w) if and only if some reduced expression of w ends with s.
Given I ⊆ S, the parabolic subgroup W I of W is generated by I. The pair (W I , I) is a Coxeter system whose Coxeter diagram has vertex set I and has labeled edges (s, t) of the Coxeter diagram of (W, S) for all s, t ∈ I. Each left coset of W I in W has a unique minimal representative. The set of all minimal representatives of left W I -cosets is W I := {w ∈ W : D(w) ⊆ S \ I}. Every element of W can be written uniquely as w = w I · I w, where w I ∈ W I and I w ∈ W I ; this implies ℓ(w) = ℓ(w I ) + ℓ( I w). Let When W is finite, the descent class of I is nonempty by Lusztig [25,Lemma 9.8] and becomes an interval [w 0 (I), w 1 (I)] under the left weak order of W by Björner and Wachs [6, Theorem 6.2]. Here w 0 (I) and w 1 (I) are the longest elements of W I and W S\I , respectively, and the left weak order is a partial ordering on W defined by setting u ≤ L w if there exists some reduced expression w = s 1 · · · s k such that s i · · · s k = u for some i.
Another important partial order on W is the Bruhat order: given u, w ∈ W, define u ≤ w if a reduced expression of u is a subword of some (or equivalently, every) reduced expression of w. When W is finite, its longest element w 0 is the unique maximum element in Bruhat order and can be characterized by the An important example of a finite Coxeter group is the symmetric group S n , and we will review its basic properties in Section 2.4. The representation theory of S n is well studied and can be extended to finite Coxeter groups of other types (see, e.g., Adin-Brenti-Roichman [1,2] and Humphreys [20, §8.10]). With that in mind, we adopt some notation below for the complex representation theory of a finite group.
The group algebra CG is semisimple and every CG-module is a direct sum of simple/irreducible CGsubmodules. There exists a complete list {S λ : λ ∈ Irr(CG)} of pair-wise nonisomorphic simple CGmodules, where Irr(CG) is in bijection with the set of conjugacy classes of G. By Schur's Lemma, the Cartan matrix of CG is the identity matrix [δ λ,µ ] λ,µ∈Irr(CG) , where δ is the Kronecker delta. The span of σ(G) := ∑ g∈G g is the trivial representation of G, whose complement in CG is spanned by the set The regular representation of G has a decomposition Here d λ be the dimension of S λ for each λ ∈ Irr(CG); in particular, There exists an integer c λ µ ≥ 0 for all λ ∈ Irr(CG) and µ ∈ Irr(CH) such that Thus the Frobenius Reciprocity holds: if λ ∈ Irr(CG) and µ ∈ Irr(CH) then The above restriction formula (2.6) implies the following lemma, which will be useful in Section 5.
Proof. Since G acts trivially on S λ , so does its subgroup H. Thus c λ µ = 0 implies S µ is trivial. 2.3. 0-Hecke algebras. Now we recall the definition and properties of the 0-Hecke algebras; see, e.g., Krob-Thibon [22], Norton [26], and Stembridge [30]. The 0-Hecke algebra H S (0) of a Coxeter system (W, S) over an arbitrary field F is the specialization of the Hecke algebra H S (q) of (W, S) at q = 0, i.e, the F-algebra generated by {π s : s ∈ S} with quadratic relations π 2 s = π s for all s ∈ S and braid relations (π s π t π s · · · ) m st = (π t π s π t · · · ) m st for all distinct s, t ∈ S, where π s := T s | q=0 . There is another generating set {π s : s ∈ S} for H S (0), where π s := π s − 1 (so that π s π s = π s π s = 0), with quadratic relations π 2 s = −π s for all s ∈ S and the same braid relations as above.
There are two F-bases {π w : w ∈ W} and {π w : w ∈ W} for H S (0), where π w := π s 1 · · · π s k and π w := π s 1 · · · π s k for any reduced expression w = s 1 · · · s k . For each w ∈ W we have Assume the Coxeter system (W, S) is finite below. Norton [26] obtained a decomposition The two equalities in (2.7) are equivalent to each other by the automorphism π i → −π i of the algebra H S (0). This gives the short derivation for the Möbius function of the Bruhat order of W by Stembridge [30].
If s ∈ S and w ∈ W then by the multiplication rule (2.8) and the relation π t π t = 0 for any t ∈ S, we have . . , k. Given I, J ⊆ S, refining the above filtration to a composition series for the cyclic module P S J (0) gives . We next study certain quotients of projective indecomposable H S (0)-modules, which will help with our study of restricted representations in Section 5. Examples in type A are given by Figure 1 in Section 2.4. Given denoting the image of a ∈ P S I in Q S I,J , we have the following F-basis for Q S I,J : Since the descent class of I in W is an interval between w 0 (I) and w 1 (I) under the left weak order, and the only element w ∈ W I with D(w) = I is w 0 (I), we have N S I,I = rad(P S I ) and Q S I,I = C S I . The general result on Q S I,J is below.
Thus if Q S I,J is projective then it must be isomorphic to P S I , which forces J = S.
Lastly, we recall from our earlier work [19, §2.3] the induction and restriction formulas for representations of 0-Hecke algebras. Let I ⊆ J ⊆ S and let w be any element of W with D(w) = I. The equalities hold in the Grothendieck groups K 0 (H S (0)) and G 0 (H S (0)), respectively. If K ⊆ S then the equalities (2.13) P S K ↓ hold in the Grothendieck groups K 0 (H J (0)) and G 0 (H J (0)), respectively, where K ↓ S J consists of certain subsets of S that can be explicitly determined by a result from our earlier work [19,Prop. 17]. Furthermore, the following two-sided duality holds for induction and restriction of 0-Hecke modules: (2.14) 2.4. The symmetric groups and 0-Hecke algebras of type A. In this subsection we summarize the representation theory of the type A Coxeter groups (i.e., symmetric groups) and 0-Hecke algebras, as well as the connections to combinatorics. We put all these in a more general framework using the notion of Grothendieck groups of a tower of algebras A * : Let M and N be finitely generated (projective) modules over A m and A n , respectively. Extending the duality between G 0 (A i ) and Bergeron and Li [4] showed that, if A * satisfies certain conditions, then with the pairing −, − , the Grothendieck groups G 0 (A * ) and K 0 (A * ) become dual graded Hopf algebras whose product and coproduct are defined by (2.15 A partition λ = [λ 1 , . . . , λ ℓ ] is a weakly decreasing sequence of positive integers λ 1 ≥ · · · ≥ λ ℓ . We use square brackets for partitions to distinguish them from compositions (defined later). The size of λ is |λ| := λ 1 + · · · + λ ℓ . The length of λ is ℓ(λ) := ℓ. We say λ is a partition of n = |λ| and write λ ⊢ n. The Grothendieck group G 0 (CS * ) = K 0 (CS * ) of the tower of algebras CS * : CS 0 ֒→ CS 1 ֒→ CS 2 ֒→ · · · is a free abelian group with a basis {S λ : λ ⊢ n, n ≥ 0}. There exist integers c λ µ,ν ≥ 0, known as the Littlewood-Richardson coefficients, for all λ |= m + n, µ ⊢ m, and ν ⊢ n such that It follows from the above formulas that, with the product ⊗ and coproduct ∆ defined in (2.15), the Grothendieck group G 0 (CS * ) becomes a self-dual graded Hopf algebra, which is isomorphic to the Hopf algebra Sym of symmetric functions via the Frobenius characteristic map defined by sending S λ to the Schur function s λ for all partitions λ. The antipode is defined by S λ → (−1) |λ| S λ t for all partitions λ, where λ t is the transpose of λ. See, e.g., Grinberg and Reiner [14, §4.4] for more details.

Structure and dimension
Let (W, S) be a Coxeter system and let F be an arbitrary field. The Hecke algebra H S (q) of (W, S) with independent parameters q := (q s : s ∈ S) ∈ F S is an F-algebra generated by {T s : s ∈ S} with • quadratic relations (T s − 1)(T s + q s ) = 0 for all s ∈ S, and • braid relations (T s T t T s · · · ) m st = (T t T s T t · · · ) m st for all distinct s, t ∈ S. Taking q s = q ∈ F for all s ∈ S in the definition of H S (q) gives the usual Hecke algebra H S (q) of (W, S) over F with a single parameter q. When F = C and q ∈ C \ {0, roots of unity}, there exists an algebra isomorphism φ : H S (q) ∼ = CW by a general deformation argument of Tits or by an explicit construction of Lusztig [24]. If one only insists q s = q t whenever m st is odd, then H S (q) becomes a Hecke algebra with unequal parameters studied by Lusztig [25].

Previous results.
In this subsection we summarize the main results of our earlier work [17] on the Hecke algebra H S (q) with q ∈ F S arbitrary. Let w ∈ W with a reduced expression w = s 1 · · · s k . Then T w := T s 1 · · · T s k is well defined, thanks to the Word Property of W [5, Theorem 3.3.1]. If s ∈ S then The set {T w : w ∈ W} always spans H S (q). This spanning set is indeed a basis if and only if H S (q) is a Hecke algebra with unequal parameters, i.e., q s = q t whenever m st is odd [17, Theorem 1.2].
For any subset R ⊆ S, we use H R (q) = H R (q| R ) to denote the Hecke algebra of the Coxeter system (W R , R) with independent parameters (q r : r ∈ R). We warn the reader that H R (q) is not necessarily isomorphic to the subalgebra of H S (q) is generated by {T r : r ∈ R} [17, §3].
The collapsed subset R ⊆ S consists of all s ∈ S connected to some other t ∈ S with q s = q t via a path in the Coxeter diagram of (W, S) whose edges all have odd weights and whose vertices (including s and t) all correspond to nonzero parameters. We have [17,Theorem 3.2] (1) T r = 1 for all r ∈ R, (2) T s / ∈ F for all s ∈ S \ R, and (3) H S (q) ∼ = H S\R (q).
Thus we may assume, without loss of generality, that H S (q) is collapse free, meaning that q s q t = 0 whenever q s = q t and m st is odd. We will keep this assumption throughout the paper.

k], then T s T t = T t T s = T s .
Lemma 3.1 played an important role in our derivation of the following results [17]. First, the algebra H S (q) is commutative if and only if the Coxeter diagram of (W, S) is simply laced and exactly one of q s and q t is zero whenever m st = 3. Next, a commutative H S (q) has a basis indexed by the independent sets in the Coxeter diagram of (W, S), which is a simple bipartite graph in this case. In particular, the dimension of H S (q) is the Fibonacci number F n+2 := F n+1 + F n with F 0 := 0 and F 1 := 1 when (W, S) is of type A n for all n ≥ 1, or the Lucas number L n := F n+1 + F n−1 when (W, S) is of affine type A n for all even n ≥ 4. We conjectured that if the Coxeter diagram of (W, S) is a simple bipartite graph then the minimum dimension of H S (q) is attained when H S (q) is commutative and verified this conjecture for type A. We also constructed a basis for H S (q) in the special case when (W, S) is simply laced.

Theorem 3.2. [17] Suppose (W, S) is simply laced and H S (q) is collapse free. Then the following statements hold.
(1) The set S decomposes into a disjoint union of subsets S 1 , . . . , S k such that the elements of each S i receive the same parameter and are connected in the Coxeter diagram of (W, S), and that if s ∈ S i , t ∈ S j , i = j, then either m st = 2 or exactly one of q s and q t is zero. (2) There is a basis for H S (q) consisting of all elements of the form T w 1 · · · T w k , where w i ∈ W S i for i = 1, . . . , k and if there exist s ∈ S i and t ∈ S j with i = j such that q s = 0, m st = 3, and s occurs in some reduced expression of w i , then w j = 1.

Example 3.3.
For H S (q) represented below, where c ∈ F \ {0}, we have a partition S = S 1 ⊔ S 2 ⊔ S 3 with S 1 of type D 4 , S 2 of type E 7 , and S 3 of type A 2 . We will compute the dimension of H S (q) later.
Example 3.4. Let (W, S) be the Coxeter system of type A n , i.e., W = S n+1 and S = {s 1 , . . . , s n }. We can view q ∈ F S as a vector (q 1 , . . . , q n ) ∈ F n whose ith component is the parameter for s i . Thus we can write H(q 1 , . . . , q n ) := H S (q). For instance, the Hecke algebra H(0, 0, 1) of the Coxeter system (W, S) of type A 3 with independent parameters (q 1 , q 2 , q 3 ) = (0, 0, 1) is generated by T 1 , T 2 , T 3 and has dimension 6 + 2 = 8 since by Theorem 3.2 it has a basis {T w T u }, where w ∈ S 3 and u ∈ S 2 satisfy the condition that if s 2 occurs in some reduced expression of w then u = 1.

New results in the simply-laced case.
In this paper we focus on the Hecke algebra H S (q) of a finite simply-laced Coxeter system (W, S) with independent parameters q ∈ F S . We may assume H S (q) is collapse free. We further assume that q 1 , . . . , q ℓ are not roots of unity to avoid technicalities. It would still be interesting to explore the case when q 1 , . . . , q ℓ are allowed to be roots of unity in the future.
Definition 3.5. Let S = S 1 ⊔ · · · ⊔ S k be the partition given by Theorem 3.2. For each i ∈ [k], we write W i := S i . There exists a partition [k] = L 0 ⊔ L 1 such that q s = 0 for all s ∈ S i , i ∈ L 0 , and that q t = 0 (we can actually assume q t = 1 by Proposition 3.9 below) for all t ∈ S j , j ∈ L 1 . Define W 0 := S 0 and Given J ⊆ L 1 and i ∈ L 0 , define W J i to be the parabolic subgroup of W i generated by S J i := {s ∈ S i : m st = 2 whenever t ∈ S j , j ∈ J}. By Lemma 3.1 and Theorem 3.2, we have the following alternative description for H S (q).
(1) The subalgebra H 0 (q) of H S (q) generated by {T s : The two subalgebras H 0 (q) and H 1 (q) commute.
(4) If s ∈ S 0 and t ∈ S 1 satisfy m st = 3 then T s T t = T t T s = T s . It follows that where J := {j ∈ L 1 : w j = 1}. By Theorem 3.2, H S (q) has a basis T w 1 · · · T w k : (w 1 , · · · , w k ) ∈ W S (q) . For each k-tuple (w 1 , . . . , w k ) ∈ W S (q), we define φ(w 1 , . . . , w k ) := {j ∈ L 1 : w j = 1}. Summing up the cardinalities of the fibers of all subsets of L 1 under the map φ gives the dimension of H S (q). Example 3.7. We revisit the algebra H S (q) in Example 3.3. We have L 0 = {2} and L 1 = {1, 3}. By Proposition 3.6 and the tables below, the dimension of H S (q) is . In earlier work [17] we gave these two formulas and also showed that, for n ≥ 0, if q is an alternating sequence in {0, 1} of length n, then H(q) is a commutative algebra whose dimension equals the Fibonacci number F n+2 := F n+1 + F n with initial terms F 0 := 0 and F 1 := 1. Now combining this with Proposition 3.6 we have, for any integers k, r ≥ 0 and n ≥ 1, This recovers a well-known identity F k+2 F r+2 + F k+1 F r+1 = F k+r+3 when n = 2, and gives the number F k+2 + (n! − 1)F k+1 = F k + n!F k+1 when r = 0, which satisfies the usual Fibonacci recurrence with initial terms 1 and n! (see OEIS [27, A022096 and A022394] for n = 3, 4). We also have Next, using the algebra isomorphism φ : CW 1 ∼ = H 1 (q) given by either Tits or Lusztig [24] together with the algebra homomorphism c : H 1 (q) → C defined by c(T t ) = 1 for all t ∈ S 1 , we show that each parameter q s ∈ C \ {0, roots of unity} of the algebra H S (q) can be assumed to be 1, without loss of generality. Proposition 3.9. Let H S (q) be the Hecke algebra over F = C of a finite simply-laced Coxeter system (W, S) with independent parameters q := (q s ∈ C \ {roots of unity} : s ∈ S). Then H S (q) is isomorphic to the algebra H S (q ′ ), where q ′ = (q ′ s : s ∈ S) is defined by Proof. Let {T s : s ∈ S} and {T ′ s : s ∈ S} be the generating sets of H S (q) and H S (q ′ ) given by the definition of the two algebras. For each s ∈ S 0 define T ′′ s := T s . For each t ∈ S 1 define T ′′ t := c t φ(t), where c t := c(φ(t)) = ±1 since φ(t) 2 = 1. 2 • For any r, t ∈ S 1 with m rt = 2, the relation between T ′ s and T ′ t is the commutativity, which is also satisfied by T ′′ r = c r φ(r) and and thus the braid relation between T ′ r and T ′ t is also satisfied by T ′′ r = c r φ(r) and T ′′ t = c t φ(t). Finally, let s ∈ S i with q s = 0 and t ∈ S j with q t = 0. Then T ′′ t = c t φ(t) lies in the subalgebra of H S (q) generated by {T r : r ∈ S j } since S j is a connected component of the Coxeter diagram of (W 1 , S 1 ) by Theorem 3.2. If m sr = 2 for all r ∈ S j then the relation between T ′ s and T ′ t is the commutativity, which is also satisfied by T ′′ s = T s and T ′′ t = c t φ(t) since T s T r = T r T s for all r ∈ S j . Otherwise by Lemma 3.1, the relation between T ′ s and

Simple and projective indecomposable modules
Let H S (q) be the Hecke algebra of a finite simply-laced Coxeter system (W, S) over the complex field F = C with independent parameters q ∈ (C \ {roots of unity}) S . In this section we construct all simple H S (q)-modules and projective indecomposable H S (q)-modules, and use them to determine the quiver and representation type of H S (q).
By Proposition 3.9, we may assume q ∈ {0, 1} S , without loss of generality. Recall that S can be partitioned into S = S 1 ⊔ · · · ⊔ S k such that the elements of each S i are connected in the Coxeter diagram and all receive the same parameter. There is also a partition [k] = L 0 ⊔ L 1 such that q s = 0 for all s ∈ S i , i ∈ L 0 , and that q t = 1 for all t ∈ S j , j ∈ L 1 .

Decomposition of the regular representation.
In this subsection we give a decomposition of the regular representation of H S (q) and obtain all simple and projective indecomposable H S (q)-modules. Definition 4.1. Let λ ∈ Irr(CW 1 ). We can write S λ = j∈L 1 S λ j where λ j ∈ Irr(CW j ) for all j ∈ L 1 . We define L λ 1 to be the set of all j ∈ L 1 such that W j acts on S λ nontrivially. Then Let I ⊆ S 0 and let S 0,λ denote the set of all s ∈ S 0 such that m st = 2 whenever t ∈ S j , j ∈ L λ 1 . Define P S I,λ := P S 0 I S λ ⊆ H S (q), where P S 0 I is identified with a submodule of H 0 S (q) ∼ = H S 0 (0) and S λ is identified with a submodule of H 1 S (q) ∼ = CW 1 . 2 Lusztig [24] gives an explicit isomorphism between H S (q) and CW; it is likely that the coefficient c t ∈ {±1} appearing in our proof can be determined using that isomorphism.

Proposition 4.2.
Suppose λ ∈ Irr(CW 1 ) and M is an H S 0,λ (0)-module. Then M ⊗ S λ becomes an H S (q)-module if we let T s act by zero for all s ∈ S 0 \ S 0,λ , by its action on M for all s ∈ S 0,λ , and by its action on S λ for all s ∈ S 1 .
Proof. One can verify the defining relations of H S (q) for the above H S (q)-action on M ⊗ S λ .

Lemma 4.3.
Let I ⊆ S 0 and λ ∈ Irr(CW 1 ). Identify S λ with a submodule of Proof. (i) If s ∈ S 0 \ S 0,λ then m st = 3 for some t ∈ S j with j ∈ L λ 1 , and it follows from Lemma 3.1 that T s S λ = 0 since S λ j ⊆ σ(W j ) ⊥ by the equation (2.5). If I ⊆ S 0,λ then π w 0 (I) = π w 0 (I)s π s for some s ∈ S 0 \ S 0,λ , and using π s S λ = T s S λ = 0 we obtain P S I,λ = H S 0 (0)π w 0 (I) π w 0 (S 0 \I) S λ = 0. (ii) Now assume I ⊆ S 0,λ . Since π s and π s act on S λ by 0 and −1, respectively, for all s ∈ S 0 \ S 0,λ , one can use the multiplication rule (2.8) of the 0-Hecke algebra to obtain This spanning set is indeed a basis, since the expansion of π w π w 0 (S 0,λ \I) σ in terms of the basis of H S (q) in Theorem 3.2 has a scalar multiple of π w as the leading term (i.e., the term with the smallest length) by Equation (2.7) and Lemma 3.1. Therefore Combining this with the definition of S 0,λ , we have the desired isomorphism P S I,λ ∼ = P S 0,λ I ⊗ S λ . If s ∈ S 0 \ S 0,λ then T s = π s annihilates the left hand side of this isomorphism and hence the right hand side as well. Proof. We can write H S (q) = H 0 (q)H 1 (q) as a sum of d λ copies of P S I,λ for all I ⊆ S 0 and all λ ∈ Irr(CW 1 ) by applying the decompositions (2.5) and (2.9) to H 1 S (q) and H 0 S (q), respectively. A summand P I,λ is nonzero if and only if (I, λ) ∈ Irr(H S (q)) by Lemma 4.3. To show this is indeed a direct sum, we compute the dimension. For each J ⊆ L 1 , the sum of the dimensions of the summands P S I,λ satisfying (I, λ) ∈ Irr(H S (q)) and L λ Summing this up over all subsets J ⊆ L 1 gives the dimension of H S (q) by Proposition 3.6. Hence the desired direct sum decomposition of H S (q) holds. Let (I, λ) ∈ Irr(H S (q)). Since P S I,λ is a direct summand of H S (q), it is projective. Lemma 4.3 implies Thus P S I,λ can be viewed as a module over the algebra H S 0,λ (0) j∈L λ 1 CW j . This module is indecomposable with top isomorphic to C S 0,λ I ⊗ S λ by Proposition 2.2 and its radical is rad P S 0,λ I ⊗ S λ by Proposition 2.1 (i), which is an H S (q)-submodule of P S I,λ with T s acting by 0 for all s ∈ S 0 \ S 0,λ and with T t acting by 1 for all t ∈ S j , j ∈ L 1 \ L λ 1 . Then Proposition 2.3 (i) implies that P S I,λ is an indecomposable H S (q)-module, and Proposition 2.3 (ii) implies that the top of P S I,λ as an H S (q)-module is isomorphic to We have (α, λ) ∈ Irr(H(0 m 1 n )) if and only if either α |= m + 1 and λ = n + 1, or α |= m, λ ⊢ n + 1, and λ = n + 1. The Cartan matrix of H(0 m 1 n ) is c α,β · δ λ,µ (α,λ),(β,µ)∈Irr(H S (q)) .
Proof. Let (I, λ) ∈ Irr(H S (q)). Among the composition factors of this H S (q)-module, the multiplicity of a simple module C S J,µ with (J, µ) ∈ Irr(H S (q)) is either zero if λ = µ, or equal to the multiplicity of C S 0,λ J among the composition factors of rad P S 0,λ I rad 2 P S 0,λ I if λ = µ. Therefore the full subquiver Q λ (q) is isomorphic to the quiver of the 0-Hecke algebra H S 0,λ (0), and there is no arrow between Q λ (q) and Q µ (q) if λ, µ ∈ Irr(CW 1 ) are distinct. By Lemma 4.7, the quivers of the 0-Hecke algebras generated by connected components of S 0,λ are all loopless, and the tensor product of these quivers gives the quiver of H S 0,λ (0). An example will be given in the end of this section, after we determine the representation type of H S (q) from its quiver. Recall that Duchamp, Hivert, and Thibon [10, §4.3] constructed the quiver of the 0-Hecke algebra H n (0) of type A n−1 and showed that H n (0) is of finite representation type if and only if n ≤ 3. In particular, the quiver of H 3 (0) consists of three connected components, two of type A 1 and one of type A 2 . Using this observation we obtain the representation type of the 0-Hecke algebra H 3 (0) ⊗ H 3 (0). Proof. The quiver of H 3 (0) ⊗ H 3 (0) is the tensor product of the quiver of H 3 (0) and itself, and thus contains a cycle of length four as a connected component. Orienting this cycle in such a way that it has no directed path of length two, one gets a quiver whose path algebra is isomorphic to a quotient of  Proof. Suppose |S i | ≥ 3 for some i ∈ L 0 . Since (W, S) is simply laced and S i is connected, there exists a subset I ⊆ S i which generates a Coxeter subsystem of type A 3 . The subalgebra of H S (q) generated by the set {T s : s ∈ I} is isomorphic to the 0-Hecke algebra H 4 (0), which is of infinite representation type by Duchamp-Hivert-Thibon [10, §4.3]. This implies that H S (q) is of infinite representation type, since every H 4 (0)-module becomes an H S (q)-module by letting all generators T s of H S (q) with s ∈ S \ I act by one and an indecomposable H 4 (0)-module is also an indecomposable H S (q)-module by Proposition 2.3 (i).
Next, assume |S i | = |S i ′ | = 2 for distinct i, i ′ ∈ L 0 . Then H 3 (0) ⊗ H 3 (0) is a quotient of H S (q). It follows from Proposition 2.4 (iii) and Lemma 4.9 that H S (q) is of infinite representation type.
Finally, assume |S i | ≤ 2 for all i ∈ L 0 with equality occurring at most once. Since H m (0) with m ≤ 2 and CW j with j ∈ L 1 are semisimple algebras, their quivers consist of isolated vertices. By Duchamp-Hivert-Thibon [10, §4.3], the quiver of H 3 (0) consists of three connected components, two of type A 1 and one of type A 2 . Thus each connected component of the quiver of the algebra is of type A 1 or A 2 by the definition of the tensor product of quivers. Since H S (q) is a quotient of the above algebra by the equation (3.2), it follows from Proposition 2.4 (iii) that H S (q) is of finite representation type.

Induction and restriction
Let H S (q) be the Hecke algebra of a finite simply-laced Coxeter system (W, S) with independent parameters q ∈ {0, 1} S , and let R ⊆ S. In this section we study the induction and restriction of representations between H R (q) = H R (q| R ) and H S (q), as there is an obvious algebra surjection from H R (q) to the subalgebra of H S (q) generated by {T s : s ∈ R} (which is not necessarily an isomorphism [17, §3]).
By induction on |R|, we may assume R = S \ {s} for some s ∈ S, without loss of generality. We distinguish two cases (q s = 0 and q s = 1) in the next two subsections. In each case our results exhibit a two-sided duality, i.e., both adjunctions (2.2) and (2.3) are true. 5.1. Case 1. In this subsection we study the case R = S \ {s} for some s ∈ S with q s = 0. One sees that R 0 = S 0 \ {s} and R 1 = S 1 . We first study induction from H R (q) to H S (q). Proposition 5.1. Suppose R = S \ {s} for some s ∈ S with q s = 0. Let (I, λ) ∈ Irr(H R (q)). Then where each H S (q)-module on the right hand side is projective indecomposable. Furthermore, if w is any element of W R 0,λ with D(w) = I, then we have the following equality in the Grothendieck group G 0 (H S (q)), where each H S (q)-module on the right hand side is simple.
Proof. By the structure (3.2) of the algebra H S (q) and Lemma 4.3, we have P R I,λ ↑ Proposition 5.3. Suppose R = S \ {s} for some s ∈ S with q s = 0. Let (I, λ) ∈ Irr(H S (q)). Then where each direct summand is a projective indecomposable H R (q)-module. Furthermore, we have where the right hand side is a simple H R (q)-module.
The same result holds if P is replaced with C.

Final remarks and questions
6.1. Hecke algebras at roots of unity. Let H n (q) be a Hecke algebra of type A n−1 over a field F of characteristic zero with a single parameter q = 0. For each λ ⊢ n, Dipper and James [9] constructed an H n (q)-module S λ (q), called the Specht module, whose dimension equals the number d λ of standard Young tableaux of shape λ. If q is not zero or a root of unity then {S λ (q) : λ ⊢ n} is a complete set of non-isomorphic simple H n (q)-modules. When q is a primitive kth root of unity, Dipper and James [9] also constructed a complete set of simple H n (q)-modules D µ (q), where µ runs through all partitions of n with at most k − 1 rows of equal length. However, these modules are not completely understood yet. In this paper we study the (complex) representation theory of the Hecke algebra H S (q) of a finite simply-laced Coxeter system (W, S) with independent parameters q ∈ (C \ {roots of unity}) S . A natural question to ask is, whether our results can be extended to the case when the parameters are allowed to be roots of unity.
6.2. Monoid algebra. The Hecke algebra H n (q) is a group algebra when q = 1 or a monoid algebra when q = 0. The representation theory of finite groups is of course well known. The representation theory of finite monoids has also been widely studied; see, e.g., Steinberg [29]. In fact, the representation theory of 0-Hecke algebras is a special case of the representation theory of J -trivial monoids studied by Denton, Hivert, Schilling, and Thiéry [8].
By Proposition 3.9, to study the Hecke algebra H S (q) with q ∈ (C \ {roots of unity}) S , we may assume q ∈ {0, 1} S , without loss of generality. Then H S (q) becomes a monoid algebra, although the underlying monoid is not J -trivial (nor R-trivial). Nevertheless, it may still be possible to recover our results via the representation theory of finite monoids and this is worth further investigation.

The Grothendieck groups of type A Hecke algebras.
For n ≥ 1 we define