PO-set paths and q -commuting minors Aaron Lauve LaCIM - UQAM S´ eminaire de combinatoire et d’informatique math´ ematique le 7 avril 2006 http://www.lacim.uqam.ca/ ∼ lauve (“Research”) lauve@lacim.uqam.ca
Outline I. Flags • what is a flag? • who cares? II. (commutative) Generic matrices • column-minor identities • homogeneous coordinate ring of the flag variety III. q -Generic matrices • row-quantum-minor identities • quantum flag variety IV. (noncommutative) Generic matrices • row-quasi-minor identities • specializations V. q -Commuting minors • “missing” relations • PO-set paths 1
I. Flags • Fix: an integer n > 1 and a vector space V = C n . • Fix: a sequence of integers λ : n ≥ λ 1 > λ 2 > · · · > λ s ≥ 0 . Definition (Flag). A flag Φ of shape λ is a chain of subspaces Φ : 0 ⊆ V 1 � V 2 � · · · � V s ⊆ V satisfying λ i = codim V i = n − dim V i . Denote the collection of all flags of shape λ by Fl ( λ ) . 2
I. Flags • Fix: an integer n > 1 and a vector space V = C n . • Fix: a sequence of integers λ : n ≥ λ 1 > λ 2 > · · · > λ s ≥ 0 . Definition (Flag). A flag Φ of shape λ is a chain of subspaces Φ : 0 ⊆ V 1 � V 2 � · · · � V s ⊆ V satisfying λ i = codim V i = n − dim V i . Denote the collection of all flags of shape λ by Fl ( λ ) . Example. Taking n = 6 and λ = (5 , 3 , 2) , Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } is a flag when v 1 , . . . , v 4 are linearly independent. 3
I. Flags • Fix: an integer n > 1 and a vector space V = C n . • Fix: a sequence of integers λ : n ≥ λ 1 > λ 2 > · · · > λ s ≥ 0 . Definition (Flag). A flag Φ of shape λ is a chain of subspaces Φ : 0 ⊆ V 1 � V 2 � · · · � V s ⊆ V satisfying λ i = codim V i = n − dim V i . Denote the collection of all flags of shape λ by Fl ( λ ) . Example. Taking n = 6 and λ = (5 , 3 , 2) , Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } is a flag when v 1 , . . . , v 4 are linearly independent. • Choose a basis B for V and express Φ as a matrix A (Φ) . . . 4
I. Flags • Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } a 11 a 12 a 13 a 14 a 15 a 16 V 1 a 21 a 22 a 23 a 24 a 25 a 26 V 2 A (Φ) = a 31 a 32 a 33 a 34 a 35 a 36 V 2 a 41 a 42 a 43 a 44 a 45 a 46 V 3 5
I. Flags • Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } a 11 a 12 a 13 a 14 a 15 a 16 V 1 a 21 a 22 a 23 a 24 a 25 a 26 V 2 A (Φ) = a 31 a 32 a 33 a 34 a 35 a 36 V 2 a 41 a 42 a 43 a 44 a 45 a 46 V 3 6
I. Flags • Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } a 11 a 12 a 13 a 14 a 15 a 16 V 1 a 21 a 22 a 23 a 24 a 25 a 26 V 2 A (Φ) = a 31 a 32 a 33 a 34 a 35 a 36 V 2 a 41 a 42 a 43 a 44 a 45 a 46 V 3 7
I. Flags • Φ : span { v 1 } ⊆ span { v 1 , v 2 , v 3 } ⊆ span { v 1 , v 2 , v 3 , v 4 } a 11 a 12 a 13 a 14 a 15 a 16 V 1 a 21 a 22 a 23 a 24 a 25 a 26 V 2 A (Φ) = a 31 a 32 a 33 a 34 a 35 a 36 V 2 a 41 a 42 a 43 a 44 a 45 a 46 V 3 ∗ ∗ ∗ ∗ • Unique up to change of basis! . . . multiplying by P λ = on the left. ∗ ∗ ∗ ∗ ∗ ∗ ∗ 8
I. Who Cares? · · · · · · a 11 a 12 a 1 n a 21 a 22 a 2 n A (Φ) = ( d = n − λ s ) . . . . . . . . . · · · · · · a d 1 a d 2 a dn Representation Theory Notice that GL n ( C ) ( and its many important subgroups ) permutes Fl ( λ ) by right-multiplication. 9
I. Who Cares? a 11 a 12 a 13 a 14 A (Φ) = a 21 a 22 a 23 a 24 Topology/Algebraic Geometry The space Fl ( λ ) is a (projective) algebraic variety and a CW-complex, described via Gaussian elimination (G.E.). Consider the case n = 4 , λ = (2) : � � G.E. Open cells: − → � � � � � � µ = (00) (10) (20) � � � � � � (11) (21) (22) 10
I. Who Cares? a 11 a 12 a 13 a 14 A (Φ) = a 21 a 22 a 23 a 24 Topology/Algebraic Geometry The space Fl ( λ ) is a (projective) algebraic variety and a CW-complex, described via Gaussian elimination. Consider the case n = 4 , λ = (2) : Schubert cells Ω µ : � � � � � � µ = (00) (10) (20) � � � � � � (11) (21) (22) Partitions µ with | λ | = 2 parts and part-size at most n − 2 = 2 . Combinatorics The classes [Ω µ ] in the cohomology ring H • ( Fl ( λ )) are Schur polynomials!! 11
II. (commutative) Generic Matrices Definition. Let X = ( x ij ) be an n × n matrix of commuting indeterminants. Call X a generic matrix (coordinate functions for a “generic point” in C n 2 ). · · · x i 1 j 1 x i 1 j 2 x i 1 j d · · · x i 2 j 1 x i 2 j 2 x i 2 j d Definition. For I, J ∈ [ n ] d , put X I,J = . . . . . . . . . . · · · x i d j 1 x i d j 2 x i d j d Let X J denote the special case I = (1 , 2 , . . . , d ) (take the first d rows of X ). Consider the column-minors of shape λ : M ( λ ) = { [ ] := det X I : n − | I | ∈ λ } . [ I ] Problem: Describe the relations R among the minors M ( λ ) . 12
II. (commutative) Generic Matrices Answer (Schur ‘01, Hodge ‘43): The minors M ( λ ) satisfy the Young symmetry relations � ( − 1) ℓ ( L \ Λ | Λ) [ [ L \ Λ] [Λ | M ] ( Y L,M ) 0 = ][ ] Λ ⊆ L | Λ | = r for any 1 ≤ r and any L, M ⊆ [ n ] with | M | + r ≤ | L | − r ∈ n − λ . Moreover, writing M ( λ ) = { I 1 , . . . I N } , if F ( Z 1 , . . . , Z N ) is a polynomial which is zero on substitution Z i �→ [ [ I i ] ] , then F is algebraically dependent on the ( Y L,M ) . [ [12] ][ [34] ] − [ [13] ][ [24] ] + [ [23] ][ [14] ] = 0 . Example: Non-example: (Sylvester’s Identity) (det A 123 , 123 ) (det A 2 , 2 ) = (det A 12 , 12 ) (det A 23 , 23 ) − (det A 12 , 23 ) (det A 23 , 12 ) . 13
III. q -Generic Matrices 2 3 a b 6 7 Definition. Call a matrix X q -generic if every 2 × 2 submatrix 5 satisfies: 6 7 6 7 6 7 c d 4 ( ← ) ba = q ab dc = q cd ( ↑ ) ca = q ac db = q bd ( ր ) cb = bc da = ad + ( q − q − 1 ) bc ( տ ) Define the quantum determinant by � ( − q ) − ℓ ( σ ) x i 1 j σ (1) x i 2 j σ (2) · · · x i d j σ ( d ) det q X I,J = σ ∈ S d for I, J ∈ [ n ] d , and let [ [ J ] ] now denote det q X J . Problem: Describe the relations R among the minors M q ( λ ) . 14
III. q -Generic Matrices Answer (Taft-Towber ‘91): The set R is (algebraically) generated by the relations below: q -Alternating: ( ∀ I ∈ [ n ] d ) ] = ( − q ) − ℓ ( σ ) [ [ [ I ] [ σI ] ] if σ “straightens” I. q -Young symmetry: ( ∀ L, M ⊆ [ n ] , r > 0) s.t. | M | + r ≤ | L | − r � ( − q ) − ℓ ( L \ Λ | Λ) [ 0 = [ L \ Λ] ][ [Λ | M ] ] . ( Y L,M ) Λ ⊂ L, | Λ | = r ( ∀ I, J � [ n ]) s.t. | J | < | I | q -Straightening: � ( − q ) + ℓ (Λ | I \ Λ) [ [ [ J ] ][ [ I ] ] = [ J | I \ Λ] ][ [Λ] ] . ( S J,I ) Λ ⊆ I, | Λ | = | J | ] − 1 ] + 1 [ [12] ][ [34] q [ [13] ][ [24] q 2 [ [23] ][ [14] ] = 0 . Example: Non-examples: (cf. Goodearl, ‘05) 15
IV. (noncommutative) Generic Matrices • Now let X be a matrix of noncommuting variables. Definition (Gelfand-Retakh ‘91). The ( ij ) -quasideterminant | X | ij is defined whenever X ij is invertible, and in that case, � � � � � � � � | X | ij = � � � � � � � � � � 16
IV. (noncommutative) Generic Matrices • Now let X be a matrix of noncommuting variables. Definition (Gelfand-Retakh ‘91). The ( ij ) -quasideterminant | X | ij is defined whenever X ij is invertible, and in that case, � � � � � � � � − 1 · | X | ij = = − · � � � � � � � � � � • 2 × 2 Example: � � � a 11 a 12 � = a 12 − a 11 a − 1 � � | A | 12 = 21 a 22 . � � a 21 a 22 � � � � 17
IV. (noncommutative) Generic Matrices • It is better to take ratios of quasideterminants as “column-minor” replacements Definition. Given an n × n matrix X and a partition λ , the column-minors are given by � � � ij ( X ) := | X i ∪ K | − 1 p K M quasi ( λ ) = di | X j ∪ K | dj � i, j ∈ [ n ] , K ⊆ [ n ] \ i, n − 1 − | K | ∈ λ � ucker Relations (Gelfand-Retakh ‘97, L. ’04): If L, M ⊆ [ n ] , i ∈ [ n ] \ M , Quasi-Pl¨ | M | ≤ | L | − 1 , then: ij ( X ) p L \ j � p M 1 = ji ( X ) . ( P i,L,M ) j ∈ L Problem: Describe the relations R satisfied by M quasi • Need to expand search to rational expressions, not just algebraic ones. • some are known. . . maybe all? 18
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