Representation type of cyclotomic quiver Hecke algebras in affine type A 1 Qi WANG Yau Mathematical Sciences Center Tsinghua  University @CNU, May 16, 2023. Outline Introduction KLR algebras Maximal weights References
0000000000 References 000 Introduction We start with Quiver Representation Theory. Quivers: A quiver representation: We may study quiver rep's algebraically or geometrically. A quiver representation:
● Algebraic viewpoint: • Geometric viewpoint: A quiver representation:
• Algebraic viewpoint : to find all indecomposable rep's. e.g., the above example has 6 indecomposable rep's:
● Geometric viewpoint: A quiver representation:
• Algebraic viewpoint : to find all indecomposable rep's. e.g., the above example has 6 indecomposable rep's: where K is a field (algebraically closed). • Geometric viewpoint: to fix all vector spaces V; and change matrices f , g . This gives an affine module variety. Algebraic Representation Theory Any (basic, connected) algebra A over K is isomorphic to a bound quiver algebra KQ/I. Algebraic Representation Theory Any (basic, connected) algebra A over K is isomorphic to a bound quiver algebra KQ/I. An algebra A is said to be • rep-finite if the number of indecomposable rep's is finite. • tame if it is not rep-finite, but all indecomposable rep's can be organized in a one-parameter family in each dimension. Otherwise, A is called wild. Algebraic Representation Theory Any (basic, connected) algebra A over K is isomorphic to a bound quiver algebra KQ/I. An algebra A is said to be • rep-finite if the number of indecomposable rep's is finite. • tame if it is not rep-finite, but all indecomposable rep's can be organized in a one-parameter family in each dimension. Otherwise, A is called wild.
Theorem (Drozd 1977)
The representation type of any algebra (over K ) is exactly one of rep-finite, tame and wild. Example: tame algebras e.g. is tame. Indecomposable rep's:
Kn In Kn Jn(X) Example: wild algebras
Impossible! to give a complete classification of indecomposable rep's for a wild algebra. Some examples related to Hecke algebras. ●  rep-finite:  e.g., Brauer tree algebras ●  tame:  e.g., Brauer graph algebras • wild: KLR algebras in affine type A Hecke algebras of type A The symmetric group Sn (= permutation group of {1,2, . . . , n}) is generated by {s; = ( i, i + 1) | 1 ≤ i ≤ n − 1} subject to
Hecke algebras of type A The symmetric group Sn (= permutation group of {1,2, . . . , n}) is generated by {s; = ( i, i + 1) | 1 ≤ i ≤ n − 1} subject to
The Iwahori-Hecke algebra H( Sn ) is the Z [q, q- 1 ]-algebra generated by { T i | 1 ≤ i ≤ n − 1} subject to
More Hecke algebras In the last fifty years, the representation theory of symmetric groups had a close connection with Lie theory via categorification. ● Hecke algebras of Coxeter groups, i.e., of type B, D, E, etc. ●  Cyclotomic Hecke algebras (a.k.a.  Ariki-Koike algebras).  See [Ariki-Koike, 1994], [Broue-Malle, 1993], and [Cherednik 1987]. ●  Cyclotomic quiver Hecke algebras (a.k.a.  Cyclotomic KLR algebras). See [Khovanov-Lauda, 2009] and [Rouquier, 2008]. Many classes of algebras arise in this process, whose representation type is completely determined, in particular, for (1) Hecke alg's in type ABD (Ariki, 2000); (2) Cyclotomic quiver Hecke alg's of level 1 in affine type ACD (Ariki-lijima-Park 2014, 2015); of level 2 in affine type A (Ariki 2017); (3) Schur/q-Schur/Borel-Schur/infinitesimal-Schur alg's (Xi 1993, Erdmann 1993, Doty-Erdmann-Martin 1999, Erdmann-Nakano 2001, etc); (4) block alg's of category O; (Futorny-Nakano-Pollack 1999, Boe-Nakano 2005, etc) Many classes of algebras arise in this process, whose representation type is completely determined, in particular, for (1) Hecke alg's in type ABD (Ariki, 2000); (2) Cyclotomic quiver Hecke alg's of level 1 in affine type ACD (Ariki-lijima-Park 2014, 2015); of level 2 in affine type A (Ariki 2017); of level k in affine type A (Ariki-Song-W. 2023); (3) Schur/q-Schur/Borel-Schur/infinitesimal-Schur alg's (Xi 1993, Erdmann 1993, Doty-Erdmann-Martin 1999, Erdmann-Nakano 2001, etc); (4) block alg's of category O; (Futomy-Nakano-Pollack 1999, Boe-Nakano 2005, etc) Lie theoretic data Let / = {0, 1, ..., ℓ} be an index set. Recall that +B(1),D(1),A(2),A(2),D(2),E(1),E(1),E(1),F(1),G(1),E(2),D(3) Lie theoretic data Let / = {0, 1, ..., ℓ} be an index set. Recall that
Set n;j := #(i → j). Lie theoretic data Let / = {0, 1, ..., ℓ} be an index set. Recall that
Let (A, P, Π, PV, ΠV) be the Cartan datum in type Α ( ¹¹ , where ● P = Dj=o ZNj ⊕ Zδ is the weight lattice; ● Π = {α; | i ∈ I} ⊂ P is the set of simple roots; ●  PV = Hom( P, Z ) is the coweight lattice; ● Π V = { h; | i ∈ I} ⊂ PV is the set of simple coroots. The null root is δ = αρ + αι + . . . + α ℓ . We have
A family of polynomials in type A Fix t ∈ K if ℓ = 1 and 0 ≠ t ∈ K if ℓ ≥ 2. For i,j ∈ I, we take Q i,j ( u, v ) ∈ K [ u, v ] such that Q i,j ( u, v ) = 0, Qi,j(u,v) = Qj,j(v,u) and if ℓ ≥ 2, Qe,o(u,v) = u + tv, Qi j (u,v) = 1 if j̸= e i,i ± 1. If ℓ = 1, we take Q 0,1( u, v ) = u 2 + tuv + v 2.
The quiver Hecke algebra R(n) associated with ( Qi.j(u, v))i.jcl is the Z-graded k-algebra generated by {e( ν ) | ν = ( ν 1 , ν 2 , . . . , ν n ) ∈ l n } , { x ; | 1 ≤ i ≤ n } , { ψ j | 1 ≤ j ≤ n − 1 } , subject to the following relations: (1) e(ν)e(ν/) = δν,νe(ν), Σνεια e(ν) = 1, χ,χ, = χ,χ, x,e(ν) = e(ν)χ,. v;e(v) = e(s;(v))v;, v;v; = v;v; if |i = j| > 1.
Cyclotomic quiver Hecke algebras Fix Λ ∈ P + .  The cyclotomic quiver Hecke algebra R N (n) w.r.t.  Λ is defined as the quotient of R(n) modulo the relation Cyclotomic quiver Hecke algebras Fix Λ ∈ P + .  The cyclotomic quiver Hecke algebra R N (n) w.r.t.  Λ is defined as the quotient of R(n) modulo the relation Here, R^(n) is a finite-dimensional symmetric algebra proved by Shan-Varagnolo-Vasserot in 2017.
Fix Λ ∈ P + .  The cyclotomic quiver Hecke algebra R N (n) w.r.t.  Λ is defined as the quotient of R(n) modulo the relation Here, R^(n) is a finite-dimensional symmetric algebra proved by Shan-Varagnolo-Vasserot in 2017.
An example Set Λ = kΛ 0, ℓ = 2. Then, l = {0,1,2} and R(3) is generated by {e(000), . . . , e(012), . . . , e(212), . . . }, {x1,x2,x3}, {ψ1,ψ2}, subject to the relations. An example Set Λ = kΛ 0, ℓ = 2. Then, l = {0,1,2} and R(3) is generated by {e(000), · · · , e(012), · · · , e(212), · · · }, {×1,×2,×3}, {ψ1,ψ2}, subject to the relations. Set β = α 1 + α 2 + α 3. Then, R λ (β) is generated by {e(012), e(021), e(102), e(120), e(201), e(210)}, {×1, ×2, ×3}, {√1, ¥2}, subject to ●  e(102) = e(120) = e(201) = e(210) = 0, x, ke(012) = x, ke(021) = 0; ● ψ1e(012) = ψ1e(021) = 0, ψ2e(012) = e(021)ψ2; ● x2e(012) = −x1e(012), x2e(021) = −tx1e(021); ● x³e(012) = tx²e(012) + (1 − t)x1x3e(012), etc. Known results on cyclotomic KLR algebras We know the representation type of cyclotomic KLR algebras in the following cases.
In this talk, we explain the representation type of R^( β ) in type A( 1 ) , for arbitrary Λ ∈ P + . Representation type of R^(β) ● Rickard's result tells us that two symmetric algebras have the same representation type if they are derived equivalent. Representation type of R^(β) ● Rickard's result tells us that two symmetric algebras have the same representation type if they are derived equivalent. ● Chuang and Rouquier's result tells us that R^( β ) and R^( β′ ) are derived equivalent if Λ — β and Λ — β′ lie in the same W-orbit of the set P(Λ) of weights of V(Λ), where W is the affine symmetric group generated by (for i ∈ I ) Representation type of R^(β) ● Rickard's result tells us that two symmetric algebras have the same representation type if they are derived equivalent. ● Chuang and Rouquier's result tells us that R^( β ) and R^( β′ ) are derived equivalent if Λ — β and Λ — β ′ lie in the same W-orbit of the set P(Λ) of weights of V(Λ), where W is the affine symmetric group generated by (for i ∈ I )
● A weight μ ∈ P(Λ) is maximal if μ + δ ξ P(Λ). We define
Kac's result tells us that the representatives of W-orbits in P(A) are given by {μ − mδ | μ ∈ max + (Λ), m ∈ Z >o}.
We briefly recall the construction in [Kim-Oh-Oh, 2020] as follows. Set Λ = a i Λ i 1 + a i 2 Λ i 2 + ··· + a i n Λ i n ∈ P + . We define
Suppose le(Λ) = k. Then,
We briefly recall the construction in [Kim-Oh-Oh, 2020] as follows.
Suppose le(Λ) = k. Then,
e.g., P ξ , 3 (Λ 0 + Λ 3 + Λ 6 ) with ℓ = 6 consists of Λ 0 + Λ 3 + Λ 6 , Λ 1 + Λ 2 + Λ 6, Λ 1 + Λ 3 + Λ 5, Λ 0 + Λ 4 + Λ 5, Λ 2 + Λ 3 + Λ 4, 2Λ 0 + Λ 2, Λ 4 + 2Λ 6, 2Λ 5 + Λ 6, Λ 0 + 2Λ 1, 2Λ 2 + Λ 5, Λ 1 + 2Λ 4, 2Λ 0 + Λ 2, 3Λ 3.
Theorem (Kim-Oh-Oh 2020) For any Λ ∈ P c l,k, there is a bijection φ Λ : max + (Λ) → P c l,k (Λ). Theorem (Kim-Oh-Oh 2020) For any Λ ∈ P c i,k, there is a bijection φ Λ : max + (Λ) → P c i,k (Λ). Remaining: to obtain the inverse φ − 1 . Theorem (Kim-Oh-Oh 2020) For any Λ ∈ P c l,k, there is a bijection φ Λ : max + (Λ) → P c l,k (Λ). Remaining: to obtain the inverse φ − 1. Recall that 〈 h i , Λ j 〉 = δ ij .  We define y i := 〈 h i , Λ − Λ ′ 〉 and Theorem (Kim-Oh-Oh 2020) For any Λ ∈ Pcl.k, there is a bijection φλ : max + (Λ) → Pcl Remaining: to obtain the inverse φ − 1
Then, we consider the linear equation AXt = Yt,. Proposition (Ariki-Song-W. 2023) ● The linear equation AX t = Y f, has a unique solution X = (xo, . . . , x e ) satisfying X E Z ξ +1 and min{x;} = 0. Proposition (Ariki-Song-W. 2023) ● The linear equation AX t = Y f, has a unique solution X = (xo, . . . , x ℓ ) satisfying
• The inverse map φ π : P ξ , κ (Λ) → max + (Λ) of φ Λ is given by
where X is the unique solution of AXt = Yt, as above. Proposition (Ariki-Song-W. 2023) ● The linear equation AX t = Y f, has a unique solution X = (xo, . . . , x e ) satisfying
• The inverse map φ π : P ξ , κ (Λ) → max + (Λ) of φ Λ is given by
where X is the unique solution of AXt = Yt, as above. Set βν := ∑ i ∈ ι ×i ː . Then,
Strategy to prove the results If Λ – β lies in the W-orbit of Ρ(Λ), then Thus, we only need to consider R^( β ) for β = β N + mδ with Λ′ ∈ P ξ k (Λ) and m ∈ Z ≥ o · Strategy to prove the results If Λ – β lies in the W-orbit of Ρ(Λ), then Λ − β ∈ {Λ − β N − mδ | Λ ′ ∈ Ρ ξ t,k (Λ), m ∈ Z ≥ o}. Thus, we only need to consider R^( β ) for β = β N + mδ with Λ′ ∈ P ξ k (Λ) and m ∈ Z ≥ o · Step 1: We show that R^( β ν + mδ ) is wild for all m ≥ 1 if β N̸= 0 and R N ( m δ ) is wild for all m ≥ 2, by using some new reduction theorems. (If R^( γ ) is not wild, we set γ ∈ NW( Λ ) ∪ {δ}.) Step 2: We determine the representation type of R^( γ ) for γ ∈ T(Λ) ∪ {δ}, via case-by-case consideration. (A systematic approach developed by Ariki and his collaborators is well applied to find the quiver presentation of R^(^).) Step 2: We determine the representation type of R^( γ ) for γ ∈ T(Λ) ∪ {δ}, via case-by-case consideration. (A systematic approach developed by Ariki and his collaborators is well applied to find the quiver presentation of R^(^).) Step 3: We show that NW(A) ⊂ T(A) via case-by-case consideration on small k (i.e., k = 3,4,5,6) and via induction on k ≥ 7. Introduction 0000000 algebras References OKOO Structure of P † k ( Λ ) (in type A(1) Recall that
e.g., P ξ , 3 (Λ 0 + Λ 3 + Λ 6 ) with ℓ = 6 consists of Λ 0 + Λ 3 + Λ 6 , Λ 1 + Λ 2 + Λ 6, Λ 1 + Λ 3 + Λ 5, Λ 0 + Λ 4 + Λ 5, Λ 2 + Λ 3 + Λ 4, etc. For any N' ∈ P ξ , κ (Λ) with k ≥ 2, we can write Λ' = Λ j + Λ j + Λ for some i,j ∈ I and Ã ∈ P †,k–2· Then, we define
Note that Λ i ; = Λ ′ if and only if j ≡ e i − 1. Definition 3.1 Let C(A) be an undirected graph, where we draw an edge between Λ′ and Λ″ if Λ″ = Λ′ ; for some i,j ∈ I with j̸= e i − 1. e.g., C(Λ 0 + Λ 3 + Λ 6 ) with ℓ = 6 is displayed as follows. Introductio We define
The unique solution of AXt = Yf, is given by min(X^ + Δj.j) = 0. e.g., 000 Definition 3.2 Let C(A) be the quiver where we set Λ′ → Λ′′ if XΛ′′ = XΛ′ + Δ i.j. We label this arrow by ( i,j ). e.g., C(Ao + A3 + A6) with ℓ = 6 is displayed as Proposition 3.3 For any N' ∈ P ξ , κ (Λ) with Λ'̸= Λ, there is a directed path from Λ to Λ′ in C (Λ). In particular, C (Λ) is a finite-connected quiver. Proposition 3.3 For any N' ∈ P ξ , κ (Λ) with Λ'̸= Λ, there is a directed path from Λ to Λ′ in C(Λ). In particular, C(Λ) is a finite-connected quiver. Proposition 3.4 Suppose Λ = Λ + Λ.  Then, there is a directed path if and only if there is a directed path Key Lemmas Lemma 3.5 Suppose that there is an arrow Λ ′ ( ˙ ˙ ˙ ) Λ ′′ in C (Λ). If R ( β Λ ′ ) is representation-infinite (resp.  wild), then so is RN( β Λי). Key Lemmas Lemma 3.5 Suppose that there is an arrow Λ ′ ( ˙ ˙ ˙ ) Λ ′′ in C (Λ). If R ( β ν ′ ) is representation-infinite (resp.  wild), then so is RN( β Λ ν ). Lemma 3.6 Write Λ = Λ + Λ. If R λ (β) is representation-infinite (resp. wild), then R^( β ) is representation-infinite (resp. wild). Rep-finite and tame sets Set io := ih, ih+1 := i1 and write Λ = mi,Λi, + ··· + mi,Λi, + mi,+1Λi,+1 + ··· + mi,Λi, Rep-finite and tame sets Set io := ih, ih+1 := i1 and write Λ = mi,Λi, + ··· + mi,Λi, + mi,+,Λi,+, + ··· + mi,Λi, For any 1 ≤ j ≤ h, we define F(Λ)o := {Λ ij,ij | m ij = 2} F(Λ)1 := { Λ ij,ij+1 | m ij = 1, m ij+1 = 1} T(^)1 := {^ij,ij+1 | mij = 1, mij+1 > 1 or mij > 1, mij+1 = 1} T(A)2 := { (^i,.i,)i,-1,i,+1 | mi, = 2, i,-1̸= e ij - 1, ij+1̸= e ij + 1 } if char K̸= 2 T(A) 3 := { (^ij,ij)ij,ij+1 or ij-1,ij | mij = 3, ij+1̸= c ij +1 or ij-1̸= c ij − 1} if char K̸= 3 T(A)4 := { (^ij,ij)ij,ij | mij = 4} if char K ≠ 2 T(Λ)5 := {(Λ ij,ij)ip,ip | m j = m j , = 2, ip̸= c ij ± 1,j̸= p } Set F(Λ) = {βν | Λ′ ∈ {Λ} ∪ F(Λ)o ∪ F(Λ)1}, T(Λ) = {βν | Λ′ ∈ ∪ı≤j≤5 T(Λ)j}. Theorem 3.7 (Ariki-Song-W. 2023) Suppose le(Λ) ≥ 3. Then, R N (β) is representation-finite if β ∈ F(Λ), tame if one of the following holds: ● β = δ, Λ = kΛ j, ℓ = 1 with t̸= ±2, ● β = δ, Λ = kΛ i, ℓ ≥ 2 with t̸= (−1)ℓ+1, ● β E T(Λ). Otherwise, it is wild. aximal weights References 000 e.g., rep-type of C(Ao + A3 + A6) with ℓ = 6 is displayed as
References 000