A (1) A Note on U q -Demazure Crystals n 1 Maggie Rahmoeller - PowerPoint PPT Presentation

Preliminaries Crystals Path Realization � � A (1) A Note on U q -Demazure Crystals n − 1 Maggie Rahmoeller Roanoke College June 4, 2018 QAA Conference � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Motivation 1990’s: Kashiwara and Lusztig developed crystal base theory This theory provides a combinatorial tool to study Lie algebra representation theory Applications arise in statistical physics, conformal field theory, differential equations, number theory, combinatorics, and algebraic geometry � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Affine Special Linear Lie Algebras We focus on the affine special linear Lie algebra: A (1) n − 1 = ˆ sl ( n , C ) = A n − 1 ⊗ C [ t , t − 1 ] ⊕ C c ⊕ C d , which has the following bracket structure: [ x ⊗ t i , y ⊗ t j ] = [ x , y ] ⊗ t i + j + tr( x , y ) i δ i + j , 0 c , [ d , x ⊗ t i ] = i ( x ⊗ t i ) , [ d , c ] = 0 . � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Quantum Affine Algebras Definition The quantum affine algebra U q ( g ) is the associative alge- bra over C ( q ) with unity generated by the elements e i , f i , q h h ∈ ˇ � � with the following relations: P 1 q 0 = 1 , q h q h ′ = q h + h ′ for h , h ′ ∈ ˇ P , 2 q h e i q − h = q α i ( h ) e i for h ∈ ˇ P , 3 q h f i q − h = q − α i ( h ) f i for h ∈ ˇ P , q si hi − q − si hi 4 e i f j − f j e i = δ ij q si − q − si , 1 − a ij � 1 − a ij � e 1 − a ij − k � ( − 1) k e j e k i = 0, for i � = j , 5 i k q si k =0 1 − a ij � 1 − a ij � f 1 − a ij − k � ( − 1) k f j f k i = 0, for i � = j . 6 i k q si k =0 � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Crystal Lattices Definition � g ( q ) � � � Define A 0 = � g ( q ) , h ( q ) ∈ C [ q ] , h (0) � = 0 to be the � h ( q ) principal ideal domain with C ( q ) as its field of quotients. Definition A free A 0 -submodule L of integrable g -module V q is a crystal lattice if 1 C ( q ) ⊗ A 0 L ∼ = V q , λ ∈ P L λ , L λ = L ∩ V q 2 L = � λ , e i ( L ) ⊆ L , ˜ 3 ˜ f i ( L ) ⊆ L for all i = 0 , 1 , . . . , n . � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Crystal Bases Definition A crystal base for V q is a pair ( L , B ) such that 1 L is a crystal lattice, 2 B is a C -basis of L / q L , 3 B = ∪ λ ∈ P B λ , B λ = B ∩ ( L λ / q L λ ), e i ( B ) ⊆ B ∪ { 0 } , ˜ 4 ˜ f i ( B ) ⊆ B ∪ { 0 } , 5 For b , b ′ ∈ B , ˜ f i b = b ′ if and only if ˜ e i b ′ = b . � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Crystal Graphs Definition Given a crystal base ( L , B ) for V q , we can define a crystal graph of V q by letting the elements of B be the set of vertices and by joining b ∈ B to b ′ ∈ B with an i -colored arrow b i → b ′ if and only if ˜ f i b = b ′ . � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Perfect Crystals Although the crystal B ( λ ) for the irreducible highest weight U q ( A (1) n − 1 )-module V q ( λ ) is infinite, it can be realized by a finite crystal called a perfect crystal . Suppose λ ( c ) = ℓ ≥ 1. By [KKMMNN], there exists a finite crystal B ℓ called a perfect crystal of level ℓ such that B ( λ ) ∼ = · · · ⊗ B ℓ ⊗ B ℓ ⊗ B ℓ . � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Demazure Modules For a dominant integral weight λ , consider the unique irreducible integrable highest weight U q ( A (1) n − 1 )-module V q ( λ ). Definition For any w ∈ W , the extremal weight space V q ( λ ) w λ is one- dimensional with basis vector u w λ , which is called the ex- tremal vector . Definition For w ∈ W and weight space V q ( λ ) w λ = C ( q ) u w λ , the � + � A (1) Demazure module is V w ( λ ) = U q u w λ , where n − 1 U q ( A (1) n − 1 ) + is the subalgebra generated by the e i ’s. � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Demazure Crystals Definition For each w ∈ W , the Demazure module V w ( λ ) has a crystal B w ( λ ), which we call the Demazure crystal . Kashiwara, in 1993, proved that the Demazure crystal is a subset of the crystal for the associated integrable highest weight module [5]. He also showed that the Demazure crystal has the following recursive property: m B w ( λ ) \ { 0 } . � ˜ w ≺ r i w ⇒ B r i w ( λ ) = f i m ≥ 0 � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Perfect Crystals For A (1) n − 1 and ℓ ≥ 1, consider the perfect crystal: n − 1 � � � m 1 , m 2 , . . . , m n − 1 , m 0 ) ∈ Z n � � B ℓ = m i = ℓ . ≥ 0 � i =0 The Kashiwara operators act on b ∈ B ℓ by the following actions: ˜ f 0 ( b ) = ( m 1 + 1 , m 2 , . . . , m n − 1 , m 0 − 1) , ˜ f i ( b ) = ( m 1 , . . . , m i − 1 , m i +1 + 1 , . . . , m n − 1 , m 0 ) . We also define the following: n − 1 � ϕ i ( b ) = m i , ϕ 0 ( b ) = m 0 , ϕ ( b ) = ϕ i ( b )Λ i i =0 n − 1 � ε i ( b ) = m i +1 , ε n ( b ) = m 0 , ε ( b ) = ε i ( b )Λ i . i =0 � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Perfect Crystals � � A (1) Perfect crystal for U q of level 2: λ = 2Λ 0 2 (2,0,0) 1 (1,1,0) 0 1 2 (0,2,0) (1,0,1) 0 1 2 (0,1,1) 0 2 (0,0,2) � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition λ = 2Λ 0 For fixed λ , let b λ be the unique (2,0,0) element of B such that 1 ϕ ( b λ ) = λ . Then set (1,1,0) 0 1 2 λ 1 = λ, λ k +1 = ε ( b λ k ) (0,2,0) (1,0,1) 0 1 b 1 = b λ , b k +1 = b λ k +1 2 (0,1,1) 0 The sequence p λ = ( · · · ⊗ b k +1 ⊗ · · · ⊗ b 2 ⊗ b 1 ) 2 (0,0,2) is called the ground-state path . Example : � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition λ = 2Λ 0 For fixed λ , let b λ be the unique (2,0,0) element of B such that 1 ϕ ( b λ ) = λ . Then set (1,1,0) 0 1 2 λ 1 = λ, λ k +1 = ε ( b λ k ) (0,2,0) (1,0,1) 0 1 b 1 = b λ , b k +1 = b λ k +1 2 (0,1,1) 0 The sequence p λ = ( · · · ⊗ b k +1 ⊗ · · · ⊗ b 2 ⊗ b 1 ) 2 (0,0,2) is called the ground-state path . Example : (0 , 0 , 2) � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition λ = 2Λ 0 For fixed λ , let b λ be the unique (2,0,0) element of B such that 1 ϕ ( b λ ) = λ . Then set (1,1,0) 0 1 2 λ 1 = λ, λ k +1 = ε ( b λ k ) (0,2,0) (1,0,1) 0 1 b 1 = b λ , b k +1 = b λ k +1 2 (0,1,1) 0 The sequence p λ = ( · · · ⊗ b k +1 ⊗ · · · ⊗ b 2 ⊗ b 1 ) 2 (0,0,2) is called the ground-state path . Example : (0 , 2 , 0) ⊗ (0 , 0 , 2) � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition λ = 2Λ 0 For fixed λ , let b λ be the unique (2,0,0) element of B such that 1 ϕ ( b λ ) = λ . Then set (1,1,0) 0 1 2 λ 1 = λ, λ k +1 = ε ( b λ k ) (0,2,0) (1,0,1) 0 1 b 1 = b λ , b k +1 = b λ k +1 2 (0,1,1) 0 The sequence p λ = ( · · · ⊗ b k +1 ⊗ · · · ⊗ b 2 ⊗ b 1 ) 2 (0,0,2) is called the ground-state path . Example : (2 , 0 , 0) ⊗ (0 , 2 , 0) ⊗ (0 , 0 , 2) � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition λ = 2Λ 0 For fixed λ , let b λ be the unique (2,0,0) element of B such that 1 ϕ ( b λ ) = λ . Then set (1,1,0) 0 1 2 λ 1 = λ, λ k +1 = ε ( b λ k ) (0,2,0) (1,0,1) 0 1 b 1 = b λ , b k +1 = b λ k +1 2 (0,1,1) 0 The sequence p λ = ( · · · ⊗ b k +1 ⊗ · · · ⊗ b 2 ⊗ b 1 ) 2 (0,0,2) is called the ground-state path . Example : · · · ⊗ (0 , 0 , 2) ⊗ (2 , 0 , 0) ⊗ (0 , 2 , 0) ⊗ (0 , 0 , 2) � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1

Preliminaries Crystals Path Realization Path Realizations Definition A sequence p = ( · · · ⊗ p ( k + 1) ⊗ p ( k ) ⊗ · · · ⊗ p (2) ⊗ p (1)) is called a λ - path if p ( k ) = b k for k ≫ 1. Example : λ -path: · · · ⊗ (0 , 0 , 2) ⊗ (2 , 0 , 0) ⊗ (1 , 1 , 0) ⊗ (0 , 1 , 1) We can use λ -paths to find a realization of the affine crystal graph of V ( λ ) and hence for the Demazure crystal. � A (1) � A Note on U q -Demazure Crystals Maggie Rahmoeller n − 1