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(e2πi/|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, Fq the finite field with q elements. A collection of facts you may or may not have forgotten: Fqn Fq

• Degree of this extension is n: Fqn ≈ (Fq)n.
• The group of units of F×

qn ⊂ Fqn is cyclic.

• F×

qn acts on Fqn by invertible Fq-linear transformations.

We conclude that there an action of F×

qn on Gk(n).

A non-standard first example

Recall that |Gk(n)| = n k

• q

= (qn − 1)(qn − q) . . . (qn − qk−1) (qk − 1)(qk − q) . . . (qk − qk−1)

A non-standard first example

Recall that |Gk(n)| = n k

• q

= (qn − 1)(qn − q) . . . (qn − qk−1) (qk − 1)(qk − q) . . . (qk − qk−1)

Theorem (Reiner–Stanton–White)

The triple

• Gk(n),

n k

• q,t

, F×

qn

• exihibts the CSP.

n k

• q,t

:= tqn − tq0 tqk − tq0 tqn − tq1 tqk − tq1 . . . tqn − tqk−1 tqk − tqk−1

A non-standard first example

Recall that |Gk(n)| = n k

• q

= (qn − 1)(qn − q) . . . (qn − qk−1) (qk − 1)(qk − q) . . . (qk − qk−1)

Theorem (Reiner–Stanton–White)

The triple

• Gk(n),

n k

• q,t

, F×

qn

• exihibts the CSP.

n k

• q,t

:= tqn − tq0 tqk − tq0 tqn − tq1 tqk − tq1 . . . tqn − tqk−1 tqk − tqk−1 Questions on this?

CSP revisited

Vic, Sen-Peng and me (BER):

• A product of cyclic groups C = C1 × · · · × Cℓ acts on X.
• X(t) ∈ Z[t1, . . . , tℓ] a polynomial.
• ωi : Ci ֒

→ C×.

CSP revisited

Vic, Sen-Peng and me (BER):

• A product of cyclic groups C = C1 × · · · × Cℓ acts on X.
• X(t) ∈ Z[t1, . . . , tℓ] a polynomial.
• ωi : Ci ֒

→ C×. (X, X(t), C) exhibits the CSP if for every c = (c1, . . . , cℓ) ∈ C, X(ω1(c1), . . . , ωℓ(cℓ)) = |X c|. and again, X(t) carries all numerical information about the action

• f C on X.

A torus action

Let α = (α1, . . . , αℓ) be a composition of n, Vα := Fqα1 ⊕ · · · ⊕ Fqαℓ ≈ Fn

q,

Tα := F×

qα1 × · · · × F× qαℓ

Tα is a torus which acts on Gk(n). What is the polynomial Xα(t) that gives a CSP triple (Gk(n), Xα(t), Tα)?

Our main result

We give a polynomial of the form Xα(t) =

• λ⊂((n−k)k)

wt(α, k; λ) 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: (Gk(n), Xα(t), Tα).

Sums over partitions

Recall, that |Gk(n)| = n k

• q

= (qn − 1)(qn − q) . . . (qn − qk−1) (qk − 1)(qk − q) . . . (qk − qk−1) =

• λ⊂(n−k)k

q|λ| A result of Reiner–Stanton: n k

• q,t

=

• λ⊂((n−k)k)

wt((n), k; λ) The form of our answer makes sense, Xα(t) =

• λ⊂((n−k)k)

wt(α, k; λ).

The weight of a partition

n = 11, k = 5, α = (4, 3, 1, 3) λ = (6, 5, 3, 3, 1)

• −

→ 1

• 1
• 1
• 1
• 1

The weight of a partition

n = 11, k = 5, α = (4, 3, 1, 3) λ = (6, 5, 3, 3, 1)

• −

→ 1

• 1
• 1
• 1
• 1

The weight of a partition

n = 11, k = 5, α = (4, 3, 1, 3) λ = (6, 5, 3, 3, 1)

• −

→ 1

• 1
• 1
• 1
• 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] = (aq − bq)/(a − b) wt(α, x) =

• [tqi(x)

r

, tqj(x)

s

] r < s [tqi(x)+j(x)

r

, tqi(x)+βr

r

] r = s wt(α; λ) =

• x∈λ

wt(α; x)

The weight of a partition, informally

[tq2

1 , tq3 1 ]

[tq

1 , tq2 1 ]

• [t1, tq2

1 ]

• [tq

2 , tq 4 ]

[t2, tq

4 ]

• [tq

2 , t4]

[t2, t4]

• Where,

[a, b] = aq − bq a − b = aq−1 + aq−1b + · · · + bq−1

Example: q = 4, Gr2(4), α = (14)

The polynomial X(t) =

λ⊂(2,2) wt((14), λ) is

Xα(t) = 1 + [t2, t3] + [t2, t4][t3, t4] + [t1, t2][t1, t3] + [t1, t2][t1, t4][t3, t4] + [t1, t3][t1, t4][t2, t3][t2, t4]. The reduction modulo tq−1

i

− 1 is a symmetric polynomial, Xα(t1, t2, t3, t4) ≡ 37 + 15s(2,1) + 10s(2,2,2) − 20s(1,1,1) General results from the theory imply that the number of orbits of T(14) is 37.

Evaluations

What are the possible number of fixed points of T(14) on Gr2(4)(F4)? F×

4 is cyclic of order 3. Let ω = e2πi/3.

(t1, t2, t3, t4) Xα(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

Tα acts on all the flag varieties G/P, G = GLn(Fq). Let W = Sn. There is a polynomial of the form Xα =

• w∈W P
• (i,j)∈Inv(w)

wt(α, k; x) for which (G/P, Xα, Tα) exhibits the cyclic sieving phenomenon. Thanks!