Pieri rules and q -characters of Hernandez-Leclerc modules for - - PowerPoint PPT Presentation

Pieri rules and q-characters of Hernandez-Leclerc modules for quantum affine sln+1

Matheus Brito

Federal University of Parana - Brazil

Joint work with Vyjayanthi Chari

June 7, 2018

Matheus Brito

Overview

Let I = {1, · · · , n} and assume that ξ : I → Z be a function satisfying ξ(i + 1) = ξ(i) ± 1, 1 ≤ i < n.

Matheus Brito

Overview

Let I = {1, · · · , n} and assume that ξ : I → Z be a function satisfying ξ(i + 1) = ξ(i) ± 1, 1 ≤ i < n. In 2012, Hernandez and Leclerc defined an interesting subcategory Fξ of finite–dimensional representations of the quantum affine algebra associated to An - a generalization of the category C1 of Leclerc’s talk - they proved that it was a monoidal category.

Matheus Brito

Overview

Let I = {1, · · · , n} and assume that ξ : I → Z be a function satisfying ξ(i + 1) = ξ(i) ± 1, 1 ≤ i < n. In 2012, Hernandez and Leclerc defined an interesting subcategory Fξ of finite–dimensional representations of the quantum affine algebra associated to An - a generalization of the category C1 of Leclerc’s talk - they proved that it was a monoidal category. In the case when ξ is alternating (ξ(i + 1) = ξ(i) + 1 = ξ(i − 1)) or the monotonic case (ξ(i + 1) = ξ(i) + 1) Hernandez and Leclerc proved that

• ne has the following picture

ξ

• Q
• Qξ
• K0(Fξ)

∼ = A(x, Qξ)

Matheus Brito

In this talk we discuss a very different proof of their result which works for an arbitrary height function (still in type A).

Matheus Brito

In this talk we discuss a very different proof of their result which works for an arbitrary height function (still in type A). Our methods allow us to give the image of an arbitrary cluster variable in K0(Fξ); equivalently we classify all the prime representations in Fξ. They turn out to be a generalization of minimal affinizations, and can be called "minimal by parts". These kind of prime representations had been studied in [CMY].

Matheus Brito

In this talk we discuss a very different proof of their result which works for an arbitrary height function (still in type A). Our methods allow us to give the image of an arbitrary cluster variable in K0(Fξ); equivalently we classify all the prime representations in Fξ. They turn out to be a generalization of minimal affinizations, and can be called "minimal by parts". These kind of prime representations had been studied in [CMY]. We also develop a recursive formula for the cluster variables which translates to giving a q–character formula for the corresponding irreducible (HL) module in terms of the fundamental modules (the initial seed) and Kirillov–Reshetikhin modules (the frozen variables).

Matheus Brito

Notation

Uq: the quantized universal enveloping algebra of ˜ sln+1 (q not root of unity) Let P+

ξ be the free monoid generated by the set

• ωi,qξ(i)±1 : i ∈ I
• .

To each element π of P+

ξ one can associate a (unique up to

isomorphism) irreducible finite–dimensional module V (π) of Uq. Let Fξ be the full subcategory of finite–dimensional representations whose irreducible constituents are the modules V (π), π ∈ P+

ξ .

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1.

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1. For 1 ≤ i < j ≤ n define elements ω(i, j) ∈ P+

ξ by,

ω(i, j) :=

• k∈[i,j]>

ωk,qξ(k)−1

• k∈[i,j]<

ωk,qξ(k)+1,

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1. For 1 ≤ i < j ≤ n define elements ω(i, j) ∈ P+

ξ by,

ω(i, j) :=

• k∈[i,j]>

ωk,qξ(k)−1

• k∈[i,j]<

ωk,qξ(k)+1,

• Example. n = 5

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1. For 1 ≤ i < j ≤ n define elements ω(i, j) ∈ P+

ξ by,

ω(i, j) :=

• k∈[i,j]>

ωk,qξ(k)−1

• k∈[i,j]<

ωk,qξ(k)+1,

• Example. n = 5

ξ(i) : 1 2 1 2 3 Q : 1 → 2 ← 3 → 4 → 5

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1. For 1 ≤ i < j ≤ n define elements ω(i, j) ∈ P+

ξ by,

ω(i, j) :=

• k∈[i,j]>

ωk,qξ(k)−1

• k∈[i,j]<

ωk,qξ(k)+1,

• Example. n = 5

ξ(i) : 1 2 1 2 3 Q : 1 → 2 ← 3 → 4 → 5 ω(1, 4) = ω1,1ω2,q3ω3,1ω4,q3, ω(1, 5) = ω1,1ω2,q3ω3,1ω5,q4.

Matheus Brito

Let Q be quiver whose vertex set is I and with edges: i → i ± 1 if ξ(i ± 1) = ξ(i) + 1. For 1 ≤ i < j ≤ n define elements ω(i, j) ∈ P+

ξ by,

ω(i, j) :=

• k∈[i,j]>

ωk,qξ(k)−1

• k∈[i,j]<

ωk,qξ(k)+1,

• Example. n = 5

ξ(i) : 1 2 1 2 3 Q : 1 → 2 ← 3 → 4 → 5 ω(1, 4) = ω1,1ω2,q3ω3,1ω4,q3, ω(1, 5) = ω1,1ω2,q3ω3,1ω5,q4. We shall call the modules associated to ωi,qξ(i)±1, ω(i, j), fi = ωi,qξ(i)+1ωi,qξ(i)−1, HL-modules of type ξ.

Matheus Brito

Main Results

Theorem

Let x = (x1, · · · , xn, f1, · · · , fn). For any height function ξ the map ι : A(x, Qξ) → K0(Fξ) given by ι(xi) =

• [ωi,qξ(i)−1],

ξ(i) = ξ(i + 1) + 1, [ωi,qξ(i)+1], ξ(i) = ξ(i + 1) − 1, , ι(fi) = [fi], is an isomorphism of algebras. Further, the image of a an arbitrary cluster variable corresponds to a prime HL–module, ι(x[αi,j]) =      [ω(i, j + 1)], j = ip+1, [ω(i, ip + 1)], i ≤ ip < j = ip+1, [ωi,ai], ip < i < j = ip+1. Suppose that α, β ∈ Φ≥−1 are such that ι(x[α]) ⊗ ι(x[β]) is reducible. Then x[α]x[β] is not a cluster monomial.

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Corollary The map ι maps cluster monomials to irreducible modules.

Matheus Brito

Corollary The map ι maps cluster monomials to irreducible modules. Example

◮ [ω(1, 4)][ω3,q2] = [ω(1, 2)][f3][ω4,q3] + [ω1,1][f2][f4] ◮ [ω(1, 4)][ω(3, 5)] = [ω(1, 5)][ω(3, 4)] + [f1][f3][f5]

ι(x[α1,3]) = [ω(1, 4)] = [ω1,1ω2,q3ω3,1ω4,q3] ι(x[α1,5]) = [ω(1, 5)] = [ω1,1ω2,q3ω3,1ω5,q4] ι(x[α1,2]) = [ω(1, 2)] = [ω1,1ω2,q3] ι(x[α3]) = [ω(3, 4)] = [ω3,1ω4,q3] ι(x[α3,5]) = [ω(3, 5)] = [ω3,1ω5,q4].

Matheus Brito

q-character formulae

Our next result gives an explicit formula for the character of the prime representations (cluster variables) in terms of fundamental representations and the KR-modules (the initial seed). For ease of notation we only state the result here in the case when ξ is alternating. Recall that in this case ω(i, k) = ωi,ξ(i)ωi+1,ξ(i+1)+2 · · · . Then, for i < k we have ι−1([ω(i, k)]) = x[αi,k] =

• r=(ri,··· ,rk)

f ri

i · · · f rk k qr i,k,

where rj ∈ {0, 1} and rj = 0 = ⇒ rj−1 = 1 = rj+1 and qr

i,k = x1−ri i−1 x1−rk k+1 k

• j=i

x1−rj−1−rj+1

j

, where we understand ri−1 = rk+1 = 1.

Matheus Brito

Methods

In A(x, Qξ)

◮ Describe a pattern for the a sequence of mutations of Qξ; ◮ Obtain a mutation formula for all cluster variables;

In K0(Fξ)

◮ Pieri rules: explicit decomposition of the tensor product of

HL-modules with fundamental modules in Fξ. Connection

◮ Compare mutation formula with Pieri rule.

Matheus Brito

The quiver Qξ

For the height function ξ we define a new quiver Qξ, with (Qξ)0 = I ∪ I′ and for (Qξ)1 the arrows at j ∈ I are as follows: If ξ(j) = ξ(j + 1) + 1 = ξ(j + 2) then j − 1 j

• (j + 1)

j′

• If ξ(j) = ξ(j + 1) + 1 = ξ(j + 2) + 2 then

j − 1 j

• (j + 1)
• j′
• (j + 1)′

Matheus Brito

Example For ξ as before we have Qξ given by 1

• 2
• 3
• 4
• 5
• 1′

2′

• 3′

4′

• 5′
• Matheus Brito

Suppose that i < j and that we have an arrow (j − 1) → j in Qξ. Let 1 = i1 < i2 < · · · < ik be the set of sinks and sources of Qξ. Then in Qξ[i, j − 1] the edges at the node j are as follows: Set a = 1 − δi,ip, c = δj,ip+1, b = δip−1+1,ip, d = δj,ip+1 if ip ≤ i < j ≤ ip+1, then (i − 1)

a

• (j − 1)

j

• c
• 1−d

(j + 1)

d

• i′

1−a

• j′

(j + 1)′

1−d

• if i < ip ≤ j ≤ ip+1, then

ip − 1

a

• (j − 1)

j

c

• 1−d

(j + 1)

d

• i′

p b

• j′

(j + 1)′

1−d

• Matheus Brito

Cluster Algebras

The non–frozen cluster variables x[α], α ∈ R+ ∪ {−αi : 1 ≤ i ≤ n} are given as

• follows. We adopt the convention that αi,k = αk if k ≤ i and αi,k = αi + · · · + αk

if k > i. Let 1 = i1 < · · · < ik = n, be the sinks and sources of Qξ x[−αi] = xi, xix[αi] = (1 − δξ(i),ξ(i+2)) (fixi+1 + fi+1xi−1) + δξ(i),ξ(i+2)(fi + xi−1xi+1).

1

if ip ≤ i < j ≤ ip+1. Then, xjx[αi,j] = f c

j x[αi,j−1]xj+1 + fj+1f 1−a i

xa

i−1,

if j < ip+1 xjx[αi,j] = f 1−a

i

xj+1xa

i−1 + f c j x[αi,j−1,

if j = ip+1.

2

if i < ip < j ≤ ip+1. xjx[αi,j] = f c

j x[αi,j−1]xj+1 + fj+1f b ipx[αi,ip−1],

if j < ip+1 xjx[αi,j] = f c

j x[αi,j−1] + f b ipxj+1x[αi,ip−1],

if j = ip+1.

Matheus Brito