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 sl n +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 F ξ of finite–dimensional representations of the quantum affine algebra associated to A n - a generalization of the category C 1 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 F ξ of finite–dimensional representations of the quantum affine algebra associated to A n - a generalization of the category C 1 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 one has the following picture ξ Q Q ξ ∼ K 0 ( F ξ ) A ( x , Q ξ ) = 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 K 0 ( F ξ ) ; equivalently we classify all the prime representations in F ξ . 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 K 0 ( F ξ ) ; equivalently we classify all the prime representations in F ξ . 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 U q : the quantized universal enveloping algebra of ˜ sl n +1 ( q not root of unity) Let P + ξ be the free monoid generated by the set � � ω i,q ξ ( 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 U q . Let F ξ 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 ) := ω k,q ξ ( k )+1 , ω k,q ξ ( k ) − 1 k ∈ [ i,j ] > k ∈ [ i,j ] < 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 ) := ω k,q ξ ( k )+1 , ω k,q ξ ( k ) − 1 k ∈ [ i,j ] > k ∈ [ i,j ] < 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 ) := ω k,q ξ ( k )+1 , ω k,q ξ ( k ) − 1 k ∈ [ i,j ] > k ∈ [ i,j ] < 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 ) := ω k,q ξ ( k )+1 , ω k,q ξ ( k ) − 1 k ∈ [ i,j ] > k ∈ [ i,j ] < Example. n = 5 ξ ( i ) : 1 2 1 2 3 Q : 1 → 2 ← 3 → 4 → 5 ω (1 , 4) = ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 4 ,q 3 , ω (1 , 5) = ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 5 ,q 4 . 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 ) := ω k,q ξ ( k )+1 , ω k,q ξ ( k ) − 1 k ∈ [ i,j ] > k ∈ [ i,j ] < Example. n = 5 ξ ( i ) : 1 2 1 2 3 Q : 1 → 2 ← 3 → 4 → 5 ω (1 , 4) = ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 4 ,q 3 , ω (1 , 5) = ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 5 ,q 4 . We shall call the modules associated to ω i,q ξ ( i ) ± 1 , ω ( i, j ) , f i = ω i,q ξ ( i )+1 ω i,q ξ ( i ) − 1 , HL-modules of type ξ . Matheus Brito

Main Results Theorem Let x = ( x 1 , · · · , x n , f 1 , · · · , f n ) . For any height function ξ the map ι : A ( x , Q ξ ) → K 0 ( F ξ ) given by � [ ω i,q ξ ( i ) − 1 ] , ξ ( i ) = ξ ( i + 1) + 1 , ι ( x i ) = , [ ω i,q ξ ( i )+1 ] , ξ ( i ) = ξ ( i + 1) − 1 , ι ( f i ) = [ f i ] , is an isomorphism of algebras. Further, the image of a an arbitrary cluster variable corresponds to a prime HL–module,  [ ω ( i, j + 1)] , j � = i p +1 ,   ι ( x [ α i,j ]) = [ ω ( i, i p + 1)] , i ≤ i p < j = i p +1 ,  [ ω i,a i ] , i p < i < j = i p +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. Matheus Brito

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)][ ω 3 ,q 2 ] = [ ω (1 , 2)][ f 3 ][ ω 4 ,q 3 ] + [ ω 1 , 1 ][ f 2 ][ f 4 ] ◮ [ ω (1 , 4)][ ω (3 , 5)] = [ ω (1 , 5)][ ω (3 , 4)] + [ f 1 ][ f 3 ][ f 5 ] ι ( x [ α 1 , 3 ]) = [ ω (1 , 4)] = [ ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 4 ,q 3 ] ι ( x [ α 1 , 5 ]) = [ ω (1 , 5)] = [ ω 1 , 1 ω 2 ,q 3 ω 3 , 1 ω 5 ,q 4 ] ι ( x [ α 1 , 2 ]) = [ ω (1 , 2)] = [ ω 1 , 1 ω 2 ,q 3 ] ι ( x [ α 3 ]) = [ ω (3 , 4)] = [ ω 3 , 1 ω 4 ,q 3 ] ι ( x [ α 3 , 5 ]) = [ ω (3 , 5)] = [ ω 3 , 1 ω 5 ,q 4 ] . 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 ) = ω i,ξ ( i ) ω i +1 ,ξ ( i +1)+2 · · · . Then, for i < k we have ι − 1 ([ ω ( i, k )]) = x [ α i,k ] = � f r i i · · · f r k k q r i,k , r =( r i , ··· ,r k ) where r j ∈ { 0 , 1 } and r j = 0 = ⇒ r j − 1 = 1 = r j +1 and k i,k = x 1 − r i i − 1 x 1 − r k � x 1 − r j − 1 − r j +1 q r , k +1 j j = i where we understand r i − 1 = r k +1 = 1 . Matheus Brito

Methods In A ( x , Q ξ ) ◮ Describe a pattern for the a sequence of mutations of Q ξ ; ◮ Obtain a mutation formula for all cluster variables; In K 0 ( F ξ ) ◮ Pieri rules: explicit decomposition of the tensor product of HL-modules with fundamental modules in F ξ . Connection ◮ Compare mutation formula with Pieri rule. Matheus Brito

� � � � � � The quiver Q ξ For the height function ξ we define a new quiver Q ξ , with ( Q ξ ) 0 = I ∪ I ′ and for ( Q ξ ) 1 the arrows at j ∈ I are as follows: If ξ ( j ) = ξ ( j + 1) + 1 = ξ ( j + 2) then � ( j + 1) j − 1 j 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 Q ξ 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 Q ξ . Let 1 = i 1 < i 2 < · · · < i k be the set of sinks and sources of Q ξ . Then in Q ξ [ i, j − 1] the edges at the node j are as follows: Set a = 1 − δ i,i p , c = δ j,i p +1 , b = δ i p − 1 +1 ,i p , d = δ j,i p +1 if i p ≤ i < j ≤ i p +1 , then a 1 − d � ( j + 1) ( i − 1) ( j − 1) j d 1 − a c 1 − d i ′ j ′ ( j + 1) ′ if i < i p ≤ j ≤ i p +1 , then a 1 − d � ( j + 1) i p − 1 ( j − 1) j d b c 1 − d i ′ j ′ ( j + 1) ′ p Matheus Brito