A Quasideterminantal Approach to Quantized Flag Varieties Aaron Lauve Rutgers University under the direction of Vladimir Retakh and Robert Lee Wilson Ph.D. Dissertation Defense April 26, 2005 http://www.rci.rutgers.edu/ ∼ lauve/thesis.html lauve@math.rutgers.edu Key Idea Proceed with Caution Watch Out!
Problem Statement Provide a means to construct noncommutative flag varieties in a variety of noncommutative settings via the quasideterminant. • Mind the Gap! • The case of Grassmannians has satisfactory results. 1
Problem Statement Provide a means to construct noncommutative flag varieties in a variety of noncommutative settings via the quasideterminant. • “It would be very important to define noncommutative flag spaces for quantum groups.” [Manin, ‘88] • Mind the Gap! • The case of Grassmannians has satisfactory results. 2
Problem Statement Provide a means to construct noncommutative flag varieties in a variety of noncommutative settings via the quasideterminant. • Traditionally, noncommutative geometry is studied by proxy: { topological spaces X } ↔ { rings of functions R ( X ) on X } • e.g. call a noncommutative algebra the “ring of functions” for some (phantom, noncommutative) variety. 3
Problem Statement Provide a means to construct noncommutative flag varieties in a variety of noncommutative settings via the quasideterminant. • In settings of “quantum group” type. . . • and only those settings possessing an “amenable determinant.” 4
Problem Statement Provide a means to construct noncommutative flag varieties in a variety of noncommutative settings via the quasideterminant. • “A main organizing tool in noncommutative algebra.” [Gelfand-G-Retakh-Wilson, ‘02] • In the commutative case, it looks like ± det A det A ij . • Has a Cramer’s Rule. • Is zero when matrix isn’t of full rank. . . . . . . . . . 5
Notation • Denote the set { 1 , 2 , . . . , n } by [ n ] . • Fix an n × n matrix A . – If i, j ∈ [ n ] then A ij denotes the deletion of row i and column j . – If I, J ⊆ [ n ] then A I,J indicates we keep only rows I and columns J . – If I ⊆ [ n ] with | I | = d , we abbreviate A I, [ d ] by A I . • Fix two sets I = { i 1 , . . . , i r } , J = { j 1 , . . . , j s } and k ∈ [ n ] \ I . – We write kI for { k } ∪ I . – We write I | J for the sequence ( i 1 , . . . , i r , j 1 , . . . , j s ) . – We write ℓ ( I | J ) for the length of the derangement I | J (the min. number of adjacent swaps needed to put I | J in increasing order). 6
Flag Varieties • Fix an integer n > 1 and a sequence γ = ( γ 1 , . . . , γ r ) of positive integers summing to n . Fix a vector space V = C n with basis B . Definition (Flags). A flag Φ of shape γ is a left coset representative of Fℓ ( γ ) := GL n ( C ) / P + γ where ∗ ... P + γ = 0 • Focus on γ = (1 , 1 , . . . , 1) for simplicity. Write Fℓ ( n ) in this case. • Another special case is γ = ( d, n − d ) . It describes the Grassmannian Gr ( d, n ) , the set of d -dimensional subspaces of V . • Fℓ ( n ) is made into a (projective) variety by the Pl¨ ucker embedding: η : A �→ { det A I | I ⊆ [ n ] , | I | = d, 1 ≤ d < n } , a map into P γ := PC ( n 1 ) × PC ( n 2 ) × · · · × PC ( n n − 1 ) . 7
Pl¨ ucker Coordinates • A point π = ( p I ) ∈ P γ belongs to η ( Fℓ ( n )) iff π satisfies: Definition (The Young Symmetry Relations ( Y L,M ) ( u ) ). Given L, M ⊆ [ n ] with | L | = s + u, | M | = t − u and s ≥ t � ( − 1) ℓ ( L \ Λ | Λ)+ ℓ (Λ | M ) p L \ Λ p Λ ∪ M . 0 = Λ ⊂ L | Λ | = u • or add alternating relations for the symbols p I and rewrite as � ( − 1) ℓ ( L \ Λ | Λ) p L \ Λ p Λ | M . 0 = Λ ⊂ L | Λ | = u • In this case, call the coordinates of π Pl¨ ucker coordinates. Theorem (Hodge-Pedoe, ‘47). A homogeneous polynomial F in the homogeneous coordinate ring C [ f I ] for P γ is zero on η if and only if it is in the ideal generated by the (right-hand sides of the) relations ( Y L,M ) ( u ) (replacing p ’s with f ’s). 8
Flag Algebra Definition (Flag Algebra). The flag algebra F ( n ) , the homogeneous coordinate ring for � f I | I ∈ [ n ] d , 1 ≤ d < n � Fℓ ( n ) , is the C -algebra with generators and relations Alternating ( A I ) : For all I ∈ [ n ] d 0 if the d elements of I are not distinct. f I = ( − 1) ℓ ( σ ) f σI if σ ∈ S d “straightens” the d -tuple I. Young symmetry ( Y L,M ) ( u ) : ( ∀ L, M ⊆ [ n ] , u > 0) s.t. | M | + u ≤ | L | − u � ( − 1) − ℓ ( L \ Λ | Λ) f L \ Λ f Λ | M . 0 = Λ ⊂ L, | Λ | = u Commuting ( C J,I ) ( ∀ I, J � [ n ]) f J f I = f J f I . 9
A q q -Deformation (“Algebra B”) q • Fix a field K q with a distinguished invertible element q . Definition (Taft-Towber, ‘91). The quantum flag algebra F q ( n ) is the K q -algebra with f I | I ∈ [ n ] d , 1 ≤ d < n � � generators and relations Alternating ( A I ) : For all I ∈ [ n ] d 0 if the d elements of I are not distinct. f I = ( − q ) − ℓ ( σ ) f σI if σ ∈ S d “straightens” the d -tuple I. Young symmetry ( Y L,M ) ( u ) : ( ∀ L, M ⊆ [ n ] , u > 0) s.t. | M | + u ≤ | L | − u � ( − q ) − ℓ ( L \ Λ | Λ) f L \ Λ f Λ | M . 0 = Λ ⊂ L, | Λ | = u q q q -Straightening ( S J,I ) ( ∀ I, J � [ n ]) s.t. | J | ≤ | I | � ( − q ) ℓ (Λ | I \ Λ) f J | I \ Λ f Λ . f J f I = Λ ⊆ I, | Λ | = | J | 10
Key Features Theorem (T-T, ‘91). The quantum flag algebra F q ( n ) satisfies • F q ( n ) reduces to F ( n ) when q → 1 . • F q ( n ) and F ( n ) are graded domains sharing the same basis and rate of growth. • F q ( n ) is a comodule algebra for the quantum groups GL q ( n ) and SL q ( n ) . . . . . . . . . . View F q ( n ) as an answer for Manin (for these particular quantum groups). After this theorem, one may safely say, the quantum flag algebra of Taft and Towber is the correct deformation for this noncommutative setting. 11
Different Approach: Noncommutative Flags • Try to deform the flags themselves, not the algebra of functions on them. • Hopefully arrive at the same algebra F q ( n ) . Preliminary Steps are Identical • Fix a skew-field D and a free D -module V = D n (must choose: left or right?) • A suitable notion of a (left/right) flag Φ exists. • A matrix representation A (Φ) exists. • A (Φ) is unique up to (left/right) multiplication by triangular matrices over D . Questions 1. Can we find a description of these flags Fℓ ( n ) in terms of coordinates? 2. Can we find a set of relations among the coordinates that characterize Fℓ ( n ) ? 12
Quasideterminants • Fix a matrix A = ( a kl ) ∈ M n ( R ) for some (noncommutative) ring R . Write A ij for the submatrix built from A by deleting row i and column j . Definition (Gelfand-Retakh, ‘91). The ( ij ) -quasideterminant | A | ij is defined whenever A ij is invertible, and in that case, � � � � � � � � | A | ij = � � � � � � � � � � ij 13
Quasideterminants • Fix a matrix A = ( a kl ) ∈ M n ( R ) for some (noncommutative) ring R . Write A ij for the submatrix built from A by deleting row i and column j . Definition (Gelfand-Retakh, ‘91). The ( ij ) -quasideterminant | A | ij is defined whenever A ij is invertible, and in that case, � � � � � � � � | A | ij = � � � � � � � � � � ij − 1 · = − · | A | 11 = a 11 − a 12 a − 1 • 2 × 2 Example: 22 a 21 . 14
Quasi-Pl¨ ucker Coordinates Definition. Given an n × n matrix A and an integer 0 < d < n , the (right) ucker coordinates of size d are given by quasi-Pl¨ � � � ij ( A ) := | A iK | is | A jK | − 1 r K � i, j ∈ [ n ] , K ⊆ [ n ] \ j, | K | = d − 1 � js ucker coordinates r K ij ( A ) satisfy Theorem (G-R, ‘97). The quasi-Pl¨ • r K ij ( A ) is independent of s (appearing in definition above) ij ( A ) for all g ∈ U + • r K ij ( A · g ) = r K n • If F ( A ) is some rational function in the a ij which is U + n -invariant, then F is a rational function in the r K ij ( A ) . • Quasi-Pl¨ ucker Relations ( P i,L,M ) : If L, M ⊆ [ n ] , i ∈ [ n ] \ M , | M | = | L | − 1 , then: r L \ j � ij ( A ) · r M 1 = ji ( A ) . j ∈ L 15
q -Generic Flags q q • Fix D and the flags Fℓ ( n ) over D . q Definition. A flag Φ is called q q -generic if there is some matrix representation A (Φ) whose entries a ij satisfy the defining relations of the quantum matrix algebra M q ( n ) built on a square matrix T . Let X q denote the set of q -generic flags of Fℓ ( n ) . Definition. There is a notion of quantum determinant det q ( - ) for T and its submatrices T J,K . We call the collection { det q A I | I ⊆ [ n ] } the (row) quantum Pl¨ ucker coordinates of A (of Φ ). • Another Key Feature of F q ( n ) : Theorem (T-T, ‘91). The quantum flag algebra F q ( n ) is isomorphic to the subalgebra of M q ( n ) generated by the quantum Pl¨ ucker coordinate functions { det q T I | I ⊆ [ n ] } for X q . 16
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