Torus sieving of finite Grassmannians Andrew Berget (joint with Jia - PowerPoint PPT Presentation

Torus sieving of finite Grassmannians Andrew Berget (joint with Jia Huang, UMN) Department of Mathematics University of California, Davis March 12, 2011

Cyclic sieving phenomenon (CSP) See the recent survey by Sagan!

Cyclic sieving phenomenon (CSP) Vic and the Dennis’: • A cyclic group C acts on a finite set X . • X ( t ) ∈ Z [ t ] a polynomial. ( X , X ( t ) , C ) exhibits the CSP if for every c ∈ C , X ( e 2 π i / | c | ) = | X c | = |{ x ∈ X : c ( x ) = x }| . X ( t ) carries all the information about the orbit structure, X (1) = | X |

A non-standard first example Let q be a prime power, F q the finite field with q elements. A collection of facts you may or may not have forgotten: F q n F q • Degree of this extension is n : F q n ≈ ( F q ) n . • The group of units of F × q n ⊂ F q n is cyclic. • F × q n acts on F q n by invertible F q -linear transformations. We conclude that there an action of F × q n on G k ( n ).

A non-standard first example Recall that = ( q n − 1)( q n − q ) . . . ( q n − q k − 1 ) � n � | G k ( n ) | = ( q k − 1)( q k − q ) . . . ( q k − q k − 1 ) k q

A non-standard first example Recall that = ( q n − 1)( q n − q ) . . . ( q n − q k − 1 ) � n � | G k ( n ) | = ( q k − 1)( q k − q ) . . . ( q k − q k − 1 ) k q Theorem (Reiner–Stanton–White) The triple � � � n � , F × G k ( n ) , q n k q , t exihibts the CSP. := t q n − t q 0 t q n − t q 1 t q k − t q 1 . . . t q n − t q k − 1 � n � t q k − t q 0 t q k − t q k − 1 k q , t

A non-standard first example Recall that = ( q n − 1)( q n − q ) . . . ( q n − q k − 1 ) � n � | G k ( n ) | = ( q k − 1)( q k − q ) . . . ( q k − q k − 1 ) k q Theorem (Reiner–Stanton–White) The triple � � � n � , F × G k ( n ) , q n k q , t exihibts the CSP. := t q n − t q 0 t q n − t q 1 t q k − t q 1 . . . t q n − t q k − 1 � n � t q k − t q 0 t q k − t q k − 1 k q , t Questions on this?

CSP revisited Vic, Sen-Peng and me (BER): • A product of cyclic groups C = C 1 × · · · × C ℓ acts on X . • X ( t ) ∈ Z [ t 1 , . . . , t ℓ ] a polynomial. → C × . • ω i : C i ֒

CSP revisited Vic, Sen-Peng and me (BER): • A product of cyclic groups C = C 1 × · · · × C ℓ acts on X . • X ( t ) ∈ Z [ t 1 , . . . , t ℓ ] a polynomial. → C × . • ω i : C i ֒ ( X , X ( t ) , C ) exhibits the CSP if for every c = ( c 1 , . . . , c ℓ ) ∈ C , X ( ω 1 ( c 1 ) , . . . , ω ℓ ( c ℓ )) = | X c | . and again, X ( t ) carries all numerical information about the action of C on X .

A torus action Let α = ( α 1 , . . . , α ℓ ) be a composition of n , V α := F q α 1 ⊕ · · · ⊕ F q αℓ ≈ F n q , T α := F × q α 1 × · · · × F × q αℓ T α is a torus which acts on G k ( n ). What is the polynomial X α ( t ) that gives a CSP triple ( G k ( n ) , X α ( t ) , T α )?

Our main result We give a polynomial of the form � X α ( t ) = wt ( α, k ; λ ) λ ⊂ (( n − k ) k ) where ◮ wt ( α, k ; λ ) is a product over the cells of λ . ◮ The weight of each cell in λ is a polynomial in two variables. ◮ The determination of wt ( α, k ; λ ) is entirely elementary. Theorem (B–Huang) The following is a CSP triple: ( G k ( n ) , X α ( t ) , T α ) .

Sums over partitions Recall, that = ( q n − 1)( q n − q ) . . . ( q n − q k − 1 ) � n � � q | λ | | G k ( n ) | = ( q k − 1)( q k − q ) . . . ( q k − q k − 1 ) = k q λ ⊂ ( n − k ) k A result of Reiner–Stanton: � n � � = wt (( n ) , k ; λ ) k q , t λ ⊂ (( n − k ) k ) The form of our answer makes sense, � X α ( t ) = wt ( α, k ; λ ) . λ ⊂ (( n − k ) k )

The weight of a partition n = 11 , k = 5 , α = (4 , 3 , 1 , 3) λ = (6 , 5 , 3 , 3 , 1) • • • • • • 1 • 0 • 0 • 0 • • 0 • • • • • • 1 • 0 • 0 • • 0 • • • • − → 1 • 0 • • 0 • • • • 1 • • 0 • • 1 •

The weight of a partition n = 11 , k = 5 , α = (4 , 3 , 1 , 3) λ = (6 , 5 , 3 , 3 , 1) • • • • • • 1 • 0 • 0 • 0 • • 0 • • • • • • 1 • 0 • 0 • • 0 • • • • − → 1 • 0 • • 0 • • • • 1 • • 0 • • 1 •

The weight of a partition n = 11 , k = 5 , α = (4 , 3 , 1 , 3) λ = (6 , 5 , 3 , 3 , 1) 1 • 0 • 0 • 0 • • 0 • • • • • • • 1 • 0 • 0 • • 0 • • • • • • 1 • 0 • • 0 • • • • − → 1 • • 0 • • • • • 1 •

The weight of a partition n = 11 , k = 5 , α = (4 , 3 , 1 , 3) λ = (6 , 5 , 3 , 3 , 1) • • • • • • • • • • • • • • • • • • • • • • • • • • • • − → • • • • • • • •

The weight of a partition, formally • • • • • • • • • • • • • • • • • • Three pieces of data for a cell x : Which block ( r , s ) it’s in, its horiztonal i ( x ) and vertical j ( x ) distance from nearest corner | . [ a , b ] = ( a q − b q ) / ( a − b ) [ t q i ( x ) , t q j ( x ) � ] r < s r s wt ( α, x ) = [ t q i ( x )+ j ( x ) , t q i ( x )+ β r ] r = s r r � wt ( α ; λ ) = wt ( α ; x ) x ∈ λ

The weight of a partition, informally [ t q 2 1 , t q 3 1 , t q 2 [ t q 1 ] 1 ] • • • • [ t 1 , t q 2 1 ] • • • • [ t q 2 , t q [ t 2 , t q • 4 ] 4 ] [ t q • 2 , t 4 ] [ t 2 , t 4 ] • Where, [ a , b ] = a q − b q = a q − 1 + a q − 1 b + · · · + b q − 1 a − b

Example: q = 4, Gr 2 (4), α = (1 4 ) λ ⊂ (2 , 2) wt ((1 4 ) , λ ) is The polynomial X ( t ) = � X α ( t ) = 1 + [ t 2 , t 3 ] + [ t 2 , t 4 ][ t 3 , t 4 ] + [ t 1 , t 2 ][ t 1 , t 3 ] + [ t 1 , t 2 ][ t 1 , t 4 ][ t 3 , t 4 ] + [ t 1 , t 3 ][ t 1 , t 4 ][ t 2 , t 3 ][ t 2 , t 4 ] . The reduction modulo t q − 1 − 1 is a symmetric polynomial, i X α ( t 1 , t 2 , t 3 , t 4 ) ≡ 37 + 15 s (2 , 1) + 10 s (2 , 2 , 2) − 20 s (1 , 1 , 1) General results from the theory imply that the number of orbits of T (1 4 ) is 37.

Evaluations What are the possible number of fixed points of T (1 4 ) on Gr 2 (4)( F 4 )? F × 4 is cyclic of order 3. Let ω = e 2 π i / 3 . ( t 1 , t 2 , t 3 , t 4 ) X α ( t ) ( ω − 1 , ω, ω, ω ) 42 ( ω − 1 , ω − 1 , ω, ω ) 27 (1 , ω − 1 , ω, ω ) 12 (1 , 1 , ω, ω ) 27 (1 , 1 , ω − 1 , ω ) 12 (1 , 1 , 1 , 1) 357

Generalizations T α acts on all the flag varieties G / P , G = GL n ( F q ). Let W = S n . There is a polynomial of the form � � X α = wt ( α, k ; x ) w ∈ W P ( i , j ) ∈ Inv( w ) for which ( G / P , X α , T α ) exhibits the cyclic sieving phenomenon. Thanks!