Global crystal bases and q-Schur algebras

Global crystal bases and q-Schur algebras 

Anna Stokke 

University of Winnipeg 

a.stokke@uwinnipeg.ca 

June 2, 2012 

Anna Stokke (University of Winnipeg) Global bases & q-Schur algebras June 2, 2012 1 / 20

The quantum enveloping algebra U q (gl n ) 

U q (gl n ) is the associative algebra over C(q) with generators 

e i , f i , 1 ≤ i < n, K j , K −1 

j 

, 1 ≤ i ≤ n 

subject to certain relations involving the indeterminate q. 




e i , f i , 1 ≤ i < n, K j , K −1 

j 

, 1 ≤ i ≤ n 





e i , f i , 1 ≤ i < n, K j , K −1 

j 

, 1 ≤ i ≤ n 


U q (sl n ) is the subalgebra generated by e i , f i , K i K −1 −1 

i+1 

and Ki K i+1 . 


Comultiplication on U q (gl n ) 

The comultiplication ∆ : U q (gl n ) ⊗ U q (gl n ) → U q (gl n ) gives an 

action on tensor products of U q (gl n )-modules: 

∆(e i ) = e i ⊗ 1 + K −1 

i 

K i+1 ⊗ e i , ∆(f i ) = f i ⊗ K i K −1 

i+1 + 1 ⊗ f i, 

∆(K i ) = K i ⊗ K i 


Young tableaux 

A sequence of non-negative integers λ = (λ 1 , λ 2 , . . . , λ n ) is a 

partition of r if λ 1 ≥ · · · ≥ λ n ≥ 0 and λ 1 + · · · + λ n = r. 





λ ⊣ r Example. (2, 2, 1) ⊣ 5 





λ ⊣ r Example. (2, 2, 1) ⊣ 5 

The Young diagram of shape λ is an arrangement of 

r = λ 1 + · · · + λ k boxes in k left-justified rows with the ith row 

consisting of λ i boxes. 





λ ⊣ r Example. (2, 2, 1) ⊣ 5 




Example. λ = (2, 2, 1) =⇒ [λ] = 





λ ⊣ r Example. (2, 2, 1) ⊣ 5 




Example. λ = (2, 2, 1) =⇒ [λ] = 

A λ-tableau is obtained by filling the boxes of the Young diagram of 

shape λ with numbers from the set {1, 2, . . . , n}. 


Semistandard Young tableaux 

T = 2 3 4 

6 6 

, S = 2 1 5 

4 5 


Semistandard Young tableaux 

T = 2 3 4 

6 6 

, S = 2 1 5 

4 5 

T is semistandard since the entries in its rows are weakly increasing 

and entries in its columns are strictly increasing. S is column 

increasing but not semistandard. 


Highest weight modules 

A U q (gl n )-module V is a highest weight module if it contains a 

highest weight vector v, where e i v = 0 for 1 ≤ i < n, such that 

U q (gl n )v = V . 





U q (gl n )v = V . 

For each partition λ with at most n nonzero parts there is a unique 

highest weight finite-dimensional irreducible U q (gl n )-module V (λ). 





U q (gl n )v = V . 

For each partition λ with at most n nonzero parts there is a unique 

highest weight finite-dimensional irreducible U q (gl n )-module V (λ). 

For χ = (χ 1 , . . . , χ n ), an n-tuple of nonnegative integers, the 

subspace V (λ) χ = {v ∈ V (λ) | K i v = q χ i 

v, i = 1, . . . , n} is a weight 

space and 

V (λ) = ⊕ χ V χ . 


Fundamental modules 

Let Λ k = (1, . . . , 1, 0, . . . , 0) ⊣ k ≤ n; can write λ = ∑ n 

i=1 a iΛ i . 




i=1 a iΛ i . 

The U q (gl n )-modules V (Λ k ) are called fundamental modules. 




i=1 a iΛ i . 

The U q (gl n )-modules V (Λ k ) are called fundamental modules. 

V (Λ k ) is an ( n 

k) 

-dimensional vector space with basis 

{[T ] | T column increasing} labeled by one-column Young tableau of 

shape Λ k = (1) k with entries from {1, 2, . . . , n}. 


Fundamental modules cont’d 

{ [T ] if i /∈ T 

K i [T ] = 

q[T ] otherwise 

{ 0 if i + 1 ∈ T or i /∈ T 

f i [T ] = 

[T ′ ] otherwise, where i is replaced with i + 1 

{ 0 if i + 1 /∈ T or i ∈ T 

e i [T ] = 

[T ′ ] otherwise, where i + 1 is replaced with i 


Fundamental modules cont’d 

{ [T ] if i /∈ T 

K i [T ] = 

q[T ] otherwise 

{ 0 if i + 1 ∈ T or i /∈ T 

f i [T ] = 

[T ′ ] otherwise, where i is replaced with i + 1 

{ 0 if i + 1 /∈ T or i ∈ T 

e i [T ] = 

[T ′ ] otherwise, where i + 1 is replaced with i 

Example. n = 3, V (Λ 2 ) has basis 

{[ ] [ ] [ ]} [ 

1 , 1 , 2 ; f 2 

2 3 3 

[ 

1 

2 

] 

is a highest weight vector. 

1 

2 

] 

= 

[ 

1 

3 

] 

, f 2 

[ 

2 

3 

] 

= 0. 


The highest weight U q (gl n )-module V (λ) 

Let λ = ∑ n 

i=1 a iΛ i be a partition into at most n parts. 





W (λ) = V (Λ n ) ⊗an ⊗ V (Λ n−1 ) ⊗a n−1 

⊗ · · · ⊗ V (Λ 1 ) ⊗a 1 

U q (gl n )-module. 

is a 





W (λ) = V (Λ n ) ⊗an ⊗ V (Λ n−1 ) ⊗a n−1 

⊗ · · · ⊗ V (Λ 1 ) ⊗a 1 


is a 

Let v λ be the tensor product of highest weight vectors of each V (Λ i ). 





W (λ) = V (Λ n ) ⊗an ⊗ V (Λ n−1 ) ⊗a n−1 

⊗ · · · ⊗ V (Λ 1 ) ⊗a 1 


is a 

Let v λ be the tensor product of highest weight vectors of each V (Λ i ). 

V (λ) = U q (gl n )v λ 


Example 

Example. n = 3, λ = (2, 1) = Λ 1 + Λ 2 , W (λ) = V (Λ 2 ) ⊗ V (Λ 1 ) 


Example 


⎧ [ ] [ ] [ ] 

⎫ 

⎨ [ ] 

[ ] 

[ ] ⎬ 

B = 1 ⊗ 

⎩ 

1 , 1 ⊗ 1 , 2 ⊗ 1 , . . . 

2 

3 

3 

⎭ 


Example 


⎧ [ ] [ ] [ ] 

⎫ 

⎨ [ ] 

[ ] 

[ ] ⎬ 

B = 1 ⊗ 

⎩ 

1 , 1 ⊗ 1 , 2 ⊗ 1 , . . . 

2 

3 

3 

⎭ 

v λ = 

[ 

1 

2 

] 

[ 

⊗ 

1 

] 

, V (λ) = U q (gl 3 )v λ . 


Example 


⎧ [ ] [ ] [ ] 

⎫ 

⎨ [ ] 

[ ] 

[ ] ⎬ 

B = 1 ⊗ 

⎩ 

1 , 1 ⊗ 1 , 2 ⊗ 1 , . . . 

2 

3 

3 

⎭ 

v λ = 

[ 

1 

2 

] 

[ 

⊗ 

1 

] 

, V (λ) = U q (gl 3 )v λ . 

Basis for W (λ) is indexed by column-increasing λ-tableaux; 

B = {ω(T ) | T column-increasing } 


Bases for V (λ) 

Dimension of V (λ) is number of semistandard tableaux of shape λ 

with entries in {1, 2, . . . , n}. 





Global crystal basis (Kashiwara, Lusztig, 1996) 






(LT-basis) Leclerc-Toffin basis (2000) 







(CL-basis) Carter-Lusztig basis (AS, 2005) 







(CL-basis) Carter-Lusztig basis (AS, 2005) 

Carter-Lusztig basis in terms of q-Schur algebra (G. Cliff, AS, 2010). 


Global crystal basis for V (λ) 

W (λ) has basis B(λ) = {ω(T ) | T column increasing } 




A = {f /h | f , h ∈ C[q], h(0) ≠ 0} 




A = {f /h | f , h ∈ C[q], h(0) ≠ 0} 

Let L W (λ) denote the A-span of B(λ) (crystal lattice of W (λ)). 




A = {f /h | f , h ∈ C[q], h(0) ≠ 0} 


Define an involution − : U q (gl n ) → U q (gl n ) by 

e i = e i , f i = f i , K j = K −1 

j 

, q = q −1 , 1 ≤ i < n, 1 ≤ j ≤ n. 




A = {f /h | f , h ∈ C[q], h(0) ≠ 0} 


Define an involution − : U q (gl n ) → U q (gl n ) by 

e i = e i , f i = f i , K j = K −1 

j 

, q = q −1 , 1 ≤ i < n, 1 ≤ j ≤ n. 

For w = uv λ ∈ V (λ), define w = uv λ , u ∈ U q (gl n ). 



Theorem. (Kashiwara) There exists a unique Q[q, q −1 ]-basis 

{G(T ) | T semistandard} of V (λ) with the properties that 

1 G(T ) ≡ ω(T ) mod qL W (λ) 

2 G(T ) = G(T ). 





1 G(T ) ≡ ω(T ) mod qL W (λ) 

2 G(T ) = G(T ). 

In general, difficult to find. 





1 G(T ) ≡ ω(T ) mod qL W (λ) 

2 G(T ) = G(T ). 

In general, difficult to find. 

Leclerc, Toffin provided an intermediate basis for V (λ), related to 

global crystal basis by an upper triangular matrix. Yields an algorithm 

for producing the global crystal basis. 


(Quantum) Carter-Lusztig basis 

Let f i,i+1 = f i and for j > i + 1, define 

f ij = f i+1,j f i − q −1 f i f i+1,j . 




Define f (k) 

i 


= f i 

k 

[k]! , where [k] = qm −q −m 

and [k]! = [k][k − 1] . . . [1]. 

q−q −1 




Define f (k) 

i 


= f i 

k 

[k]! , where [k] = qm −q −m 


q−q −1 

For T a semistandard λ-tableau, define 

∏ 

F T = 

f (γ ij ) 

ij 

= f (γ 12) 

12 

F (γ 13) 

13 

. . . f (γ 1k) 

1k 

f (γ 23) 

23 

. . . f (γ 2k) 

2k 

1≤i



Define f (k) 

i 


= f i 

k 

[k]! , where [k] = qm −q −m 


q−q −1 

For T a semistandard λ-tableau, define 

∏ 

F T = 

f (γ ij ) 

ij 

= f (γ 12) 

12 

F (γ 13) 

13 

. . . f (γ 1k) 

1k 

f (γ 23) 

23 

. . . f (γ 2k) 

2k 

1≤i

CL-basis 

Theorem (AS) The set {F T v λ | T semistandard } is a basis for V (λ). 


CL-basis 


Example. Let λ = (2, 1), n = 3. 


CL-basis 


Example. Let λ = (2, 1), n = 3. 

V (λ) has one 2-dimensional weight space; Any basis for the weight 

space is indexed by tableaux T 1 = 1 3 

2 

and T 2 = 1 2 

3 

. 


CL-basis 


Example. Let λ = (2, 1), n = 3. 

V (λ) has one 2-dimensional weight space; Any basis for the weight 

space is indexed by tableaux T 1 = 1 3 

2 

and T 2 = 1 2 

3 

. 

CL-basis is {F T1 v λ , F T2 v λ } = {f 13 v λ = (f 2 f 1 − q −1 f 1 f 2 )v λ , f 1 f 2 v λ } 

which is not the same as the global basis. 


Example cont’d 

Replace F T1 v λ with F T1 v λ + q −1 F T2 v λ = f 2 f 1 v λ . Then f 2 f 1 v λ = f 2 f 1 v λ . 




[ 

f 2 f 1 v λ = 

[ 

f 1 f 2 v λ = 

1 

2 

1 

3 

] 

] 

[ 

⊗ 

[ 

⊗ 

3 

2 

[ 

] 

+ q 

[ 

] 

+ q 

1 

3 

2 

3 

] 

] 

[ 

⊗ 

[ 

⊗ 

2 

1 

] 

, 

] 

. 




[ 

f 2 f 1 v λ = 

[ 

f 1 f 2 v λ = 

1 

2 

1 

3 

] 

] 

[ 

⊗ 

[ 

⊗ 

3 

2 

[ 

] 

+ q 

[ 

] 

+ q 

1 

3 

2 

3 

] 

] 

[ 

⊗ 

[ 

⊗ 

2 

1 

] 

, 

] 

. 

{f 2 f 1 v λ , f 1 f 2 v λ } is the global crystal basis. 


Definition of C q [x ij | 1 ≤ i, j ≤ n] 


Definition of C q [x ij | 1 ≤ i, j ≤ n] 

Define C q [x ij | 1 ≤ i, j ≤ n] to be the associative C-algebra generated 

by x ij , 1 ≤ i, j ≤ n subject to the relations: 

x il x ik = qx ik x il 

1 ≤ k < l ≤ n 

x jk x ik = qx ik x jk 

1 ≤ i < j ≤ n 

x il x jk = x jk x il 1 ≤ i < j ≤ n, 

1 ≤ k < l ≤ n 

x ik x jl − x jl x ik = (q −1 − q)x il x jk 1 ≤ i < j ≤ n, 

1 ≤ k < l ≤ n. 


q-Schur algebras 

A q (n, r) is the subspace of C q [x ij ] consisting of homogeneous 

polynomials of degree r, r ≥ 0. 





Example. x 2 11 + x 22x 13 ∈ A q (3, 2) 





Example. x 2 11 + x 22x 13 ∈ A q (3, 2) 

The dual of A q (n, r) is an algebra, called the q-Schur algebra. 

S q (n, r) = (A q (n, r)) ∗ = {ξ : A q (n, r) → C | ξ linear}. 





Example. x 2 11 + x 22x 13 ∈ A q (3, 2) 



W (λ) is an S q (n, r)-module; V (λ) = {ξv λ | ξ ∈ S q (n, r)}. 





Example. x 2 11 + x 22x 13 ∈ A q (3, 2) 



W (λ) is an S q (n, r)-module; V (λ) = {ξv λ | ξ ∈ S q (n, r)}. 

(Cliff, A.S., 2010) B 1 = {ξ T w λ | T semistandard} is a basis for V (λ). 


Relationships between bases 

Let {A(U)v λ | U semistandard} denote the LT-basis for V (λ). 




Theorem (A.S.) Let U be a semistandard λ-tableau and suppose 

that A(U)v λ = ∑ T a T F T v λ . Then 

1 a U = q k , k ∈ Z 

2 if a U ≠ 0, then U ≤ T in the lexicographic column ordering. 






1 a U = q k , k ∈ Z 


Corollary CL-basis is related to global basis by an upper triangular 

invertible matrix. 






1 a U = q k , k ∈ Z 


Corollary CL-basis is related to global basis by an upper triangular 

invertible matrix. 

Also...algorithm for producing global basis elements in terms of the 

q-Schur algebra. 


The end 

Thank you!

Global crystal bases and q-Schur algebras

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