Representation type of cyclotomic quiver Hecke algebras in affine type A 1 Qi WANG Yau Mathematical Sciences Center Tsinghua  University The 16th National Conference on Algebra @Quanzhou, November 17, 2023. Outline Introduction KLR algebras Maximal weights References Introduction KERTalgebras Maximal weights 000000000 References 000 Introduction Goal of Algebraic Representation Theory Classify all indecomposable modules of a given algebra A and all morphisms between them, up to isomorphism. Any (basic, connected) algebra A over an algebraically closed field K is isomorphic to a bound quiver algebra KQ/I.

Theorem (Drozd 1977)
The representation type of any algebra (over K ) is exactly one of rep-finite, tame and wild. An algebra A is said to be •  rep-finite if the number of indecomposable modules is finite. •  tame if it is not rep-finite, but all indecomposable modules can be organized in a one-parameter family in each dimension. Otherwise, A is called wild. "The representation type of symmetric algebras is preserved under derived equivalence."
Main Theorem (Ariki-Song-W. 2023) Suppose |Λ ≥ 3. The cyclotomic quiver Hecke algebra R^(β) of type A(1) is rep-finite if β ∈ F(Λ), tame if one of the following holds: ● β = δ, Λ = kΛj, ℓ = 1 with t̸= ±2, ● β = δ, Λ = kΛj, ℓ ≥ 2 with t̸= (−1)ℓ+1, ● β ∈ T(Λ). Otherwise, it is wild. More on 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 A, B, D, etc. ●  Cyclotomic Hecke algebras (a.k.a.  Ariki-Koike algebras).  See [Ariki-Koike, 1994], [Broue-Malle, 1993], [Cherednik 1987]. ●  Cyclotomic quiver Hecke algebras (a.k.a.  Cyclotomic KLR algebras).  See [Khovanov-Lauda, 2009],  [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 A, B (Erdmann-Nakano 2001, Ariki-Mathas 2004); (2) Cyclotomic quiver Hecke alg's of level 1 in affine type A, C, D (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; (Futomy-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 A, B (Erdmann-Nakano 2001, Ariki-Mathas 2004); (2) Cyclotomic quiver Hecke alg's of level 1 in affine type A, C, D (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) Cyclotomic quiver Hecke algebras Maximal weig
Let / = {0, 1, ..., ℓ} be an index set.  Recall that +B '', , 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 nij := #(i → j). Lie theoretic data Let / = {0,1,...,ℓ} be an index set. Recall that
Set nij := #(i → j).  We define the Cartan matrix A = (aij)i.jel by
Let (A, P, Π, PV, ΠV) be the Cartan datum in type X(1), where ● P = ZAo ⊕ ZA1 ⊕ · · ⊕ ZNe ⊕ Zo is the weight lattice; ●  Π = {α; | 0 ≤ i ≤ l} ⊂  P is the set of simple roots; ● PV = Hom(P,Z) is the coweight lattice; ●  ΠV = {h; | 0 ≤ i ≤ l}  ⊂ PV is the set of simple coroots. We have
The null root is δ, e.g.,
The quiver Hecke algebra R(n) associated with (Q;j(u,v))i,j€l is the K-algebra generated by subject to the following relations: | e(ν)e(ν ) = δν,ν,ε e(ν), Σνε ιν e(ν) = 1, χ;x; = χ;x;, χ;e(ν) = e(ν)x;. ψ;e(ν) = e(s;(ιν))ψ;, ψ;φ;j = ψ;e(; if |i − j| > 1.
A family of polynomials in affine 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 j ≠ ℓ+1 i,i ± 1. If ℓ = 1, we take Q0,1(u, v) = u² + tuv + v². Cyclotomic quiver Hecke algebras Set Λ = aoΛo + a1Λ1 + · · + a ℓ Λℓ ∈ P + , ai ∈ Z ≥o. The cyclotomic quiver Hecke algebra R^(n) is defined as the quotient of R(n) modulo the relation Cyclotomic quiver Hecke algebras Set
The cyclotomic quiver Hecke algebra R^(n) is defined as the quotient of R(n) modulo the relation Set
● R^(β) is a symmetric algebra, see [Shan-Varagnolo-Vasserot, 2017].
● R^(β) is a symmetric algebra, see [Shan-Varagnolo-Vasserot, 2017].
which is the W-orbit of the set P(A) of weights of V(A), where W is the affine symmetric group and V(A) is the integrable highest weight module of the quantum group. See: [Chuang-Rouquier, 2008].
● R^(β) is a symmetric algebra, see [Shan-Varagnolo-Vasserot, 2017].
which is the W-orbit of the set P(A) of weights of V(A), where W is the affine symmetric group and V(A) is the integrable highest weight module of the quantum group. See: [Chuang-Rouquier, 2008]. ●  A weight μ ∈ P(Λ) is maximal if μ + δ̸ξ P(Λ). We define
max†(V) Theorem (Kim-Oh-Oh 2020) There is a bijection φλ : max+(Λ) → Pξ (Λ).
In type A(1), we have
In type C (1), we have
Theorem (Ariki-Song-W. 2023)
where X = (xo,x1, . . , x e ) is the unique solution of AX t = Y f, satisfying
Introduction 0000000 KLR algebras pooopoop hall weigh References 000 Structure of P  (Λ) Maximal weig KLR algebras 000000000
A finite connected quiver
e.g., P + Λ o + Λ 3 + Λ 6 ) in type Α (1) consists of Λ o + 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.
e.g., We define
The unique solution of AXt = Yt, is given by min(X^/ + Δi,j) = 0. e.g., 000 References Rule to draw arrows Let Δ ‡  be the set of positive roots of the root system of type X. Then, the set Δ t ∪ (δ — Δ tn) gives all arrows Λ ′ —  ∧".
Key  Lemmas Lemma 1 The quiver C (Λ) of P ξ (Λ) is a finite connected quiver. Key Lemmas Lemma 1 The quiver C ( Λ ) of P ξ ( Λ ) is a finite connected quiver. Lemma 2 Suppose Λ =  λ +  λ.  There is a directed path
if and only if there is a directed path
Lemma 3 Suppose that there is an arrow Λ′ —→ Λ′′ in C̃(Λ). If R^(βλ′) is representation-infinite (resp. wild), then so is R^( β ν ). Lemma 3 Suppose that there is an arrow Λ′ —→ Λ′′ in C̃(Λ). If R^(βν′) is representation-infinite (resp. wild), then so is R^( β ν ). Lemma 4 Write Λ = ̃λ +̃λ.  If R̃λ(β) is representation-infinite (resp.  wild), then R^( β ) is representation-infinite (resp. wild). Rep-finite and tame sets in affine type A Set io := ih, ih+1 := i1 and write Λ = m i Λ i  1 + · ·· + m i Λ i  ,+ m i + 1 λ i +1 + ··· + m i λ i b Rep-finite and tame sets in affine type A Set io := ih, ih+1 := i1 and write Λ = m i Λ i + · · + m i Λ i + m ij+1 √ ij+1 + ··· + m i ⁄ n ⁄ n ⁄ n For any 1 ≤ j ≤ h, we define F(Λ)o := { Λ i ,i | m i = 2} F(A)1 := { N ij,ij+1 | m ij = 1, m ij+1 = 1} T(A)1 := { N j , j +1 | m j = 1, m j +1 > 1 or m j > 1, m j +1 = 1} T(A)2 := { (Λ j, j, ) i − 1, j +1 | m j = 2, j − 1̸= c  ij − 1, j +1̸= c ij +1 } if char K̸= 2 T(A) 3 := { (A j, j, )j,,j,+1 or j,-1,j | m j, = 3, ij+1 f e ij +1 or j,-1 f e ij -1 f char K f 3 T(A)4 := { (Λiγ,iγ)iγ,iγ | mij = 4} if char K̸= 2 T ( Λ ) 5 := { ( Λ i, i, i, )i, i, = mi, = 2, ip̸= 2, ip̸= 1, j̸= p̸= p̸Set
Theorem (Ariki-Song-W. 2023) Suppose le(Λ) ≥ 3. Then, R^(β) is representation-finite if β ∈ F(Λ), tame if one of the following holds: ● β = δ, Λ = kΛj, ℓ = 1 with t̸= ±2, ● β = δ, Λ = kΛj, ℓ ≥ 2 with t̸= (−1)ℓ+1, ● β ∈ T(Λ). Otherwise, it is wild. e.g., rep-type of C ( Λ o + Λ 3 + Λ 6 ) in type Α ( 1 is displayed as