permutation statistics in the affine Coxeter group of type A Eli - - PowerPoint PPT Presentation

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

permutation statistics in the affine Coxeter group of type A

Eli Bagno Jerusalem College of Technology Riccardo Biagioli University of Lyon 1 Robert Schwarz Bar Ilan University

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

The length function on Sn

For σ ∈ Sn (symmetric group on n letters) the length of σ (ℓ(σ)) is defined by min{r ∈ N | σ = si1 · · · sir for some i1, . . . , ir ∈ [n − 1]} For example σ = 3142 = s2s3s1, ℓ(σ) = 3.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

Statistics on Sn

The length is equal to the number of inversions inv(σ) := |{(i, j) | i < j, σ(i) > σ(j)}| The descent set of σ := σ(1) · · · σ(n) ∈ Sn is Des(σ) := {i ∈ [n − 1] | σ(i) > σ(i + 1)}, and des(σ) := | Des(σ)| is the descent number. The major index of σ ∈ Sn is maj(σ) :=

• i∈Des(σ)

i

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

Example

• Ex. If σ = 623514 ∈ S6 then

inv(σ) = 9; Des(σ) = {1, 4} des(σ) = 2 and maj(σ) = 1 + 4 = 5.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

Equidistribution over Sn

Theorem [MacMahon, 1915]

• β∈Sn

qmaj(β) =

• β∈Sn

qℓ(β) =

n−1

• i=0

(1 + q + . . . + qi). A statistic equidistributed with the length is said to be Mahonian

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition

Is there any analogue for affine Coxeter groups? Since these groups are infinite, there is very little permutation statistics on them in the literature. Couple of works, by A. Bjiorner, F . Brenti, V.Reiner‘. Recently, Clark and Ehrenborg introduced an excedance statistics for the affine Coxeter group of type A. They used enumeration of lattice points of a skew version of the root polytope of type A.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The Coxeter affine group of type A

˜ Sn is the group of all bijections π : Z → Z satisfying the following two identities: π(x + n) = π(x) + n,

n

• x=1

π(x) = n + 1 2

• with composition as group operation.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

Integer notation

The values of π on {1, . . . , n} determine the other values so we write π = [π(1), . . . , π(n)] = [π1, . . . , πn]. Example: n = 5 We represent the affine permutation π = . . . −1 1 2 3 4 5 6 7 −7 12 1 −1 −2 17 6 4 . . . by π = [1, −1, 0, −2, 17] ∈ ˜ S5

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

Colored notation

For p ∈ Z, write p = c · n + a with a ∈ {1, . . . , n} and denote p = ac. So π = ac1

1 ac2 2 . . . acn n ,

where πi = ci · n + ai. Denote also: |π| = a1 · · · an ∈ Sn.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

Example: n = 5, π = [1, −1, 0, −2, 17] = 104−15−13−123 −2 = (−1) · 5 + 3 |π| = 14532 ∈ S5

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

Define: csum(π) = csum(ac1

1 . . . acn n ) = n

• i=1

ci. Note that for each π ∈ ˜ Sn, csum(π) = 0. Hence, the group ˜ Sn can be seen as the subgroup of the wreath product Z ≀ Sn = Zn ⋊ Sn: ˜ Sn = {π ∈ Z ≀ Sn | csum(π) = 0} The numbers r1, . . . , rn are called the colors of π.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The coxeter generators

The affine group ˜ Sn is a Coxeter group Generators: S = {s0, s1, . . . , sn−1} Relations:

s2

i = 1, (i ∈ {0, . . . , n − 1})

sisi+1si = si+1sisi+1, (0 ≤ i < n) sisj = sjsi, (0 ≤ i < j < n, |j − i| > 1)

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The length generating function

The length of an element π ∈ ˜ Sn: ℓ(π) = min{r ∈ N : π = si1 · · · sir , for some i1, ..., ir ∈ [0, n − 1]}. Theorem: (Bjorner, Brenti, Shi) ℓ(π) =

• 1≤i<j≤n
• π(j) − π(i)

n

• .

The generating function of the length is equal to

• π∈˜

Sn

qℓ(π) = [n]q (1 − q)n−1 , where [n]q = 1−qn

1−q .

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The O.G.S. decomposition

G is a group. Let S = {t0, . . . , tn−1} ⊂ G s.t. each g ∈ G can be written uniquely in the form g = tkn−1

n−1 · · · tk1 1 tk0

where For torsion free generators: ki ∈ Z. For generators of order ui: 0 ≤ ki ≤ mi − 1 for some mi|ui. S is called an Ordered Generating System (O.G.S) We say that G has an O.G.S decomposition.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The O.G.S. decomposition

Clearly, every finite solvable group has an O.G.S given by its cyclic decomposition factors. Theorem:(Shwartz) The simple groups An (alternating), PSL(n, Q), PSp(2n, Q) (symplectic) and all the 5 sporadic Mathieu groups have O.G.S. decomposition. The symmetric group and the colored permutation groups also have an O.G.S decomposition.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

An O.G.S decomposition for ˜ Sn

We present an O.G.S decomposition for ˜ Sn: t1 = 1−123 . . . n1 t2 = 2−113 . . . n1 . . . tn−1 = (n − 1)−112 . . . (n − 2)n1 t0 = n−112 . . . (n − 1)1. Define for each π = tkn−1

n−1 · · · tk0 0 ,

α(π) =

n−1

• i=0

ki.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The action of t0 in multiplying from the right

Right shift all the places. Subtract 1 color from the first place. Add 1 color to the last place. Example: 112−2314050 t0 → 5−1112−23141.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The action of t±1

1

in multiplying from the right

t1

Subtract 1 color from the first place. Add 1 color to the last place. Example (112−2314050)

t1

→ (102−2314051).

t−1

1

Add 1 color to the first place. Subtract 1 color from the last place. Example (512−2314050)

t−1

1

→ (122−231405−1).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition The Coxeter genrators The O.G.S. decomposition

The action of the ti , (i > 1) in multiplying from the right

ti:

Right shift the first i places. Subtract 1 color from the first place. Add one color to the last place. All other places are staying fixed. ar1

1 ar2 2 · · · ari i · · · arn n → ari−1 i

ar1

1 · · · ari−1 i−1 · · · arn+1 n

.

t−1

i

:

Left shift the first i places. Add 1 color to the i-th place. subtract one color from the last place. All other places are staying fixed. ar1

1 ar2 2 · · · ari i · · · arn n → ar2 2 ar3 3 · · · ari i ar1+1 1

· · · arn−1

n

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

An algorithm for the O.G.S. decomposition

Let π ∈ ˜ Sn. First, apply t0 on π from the right in order to place n in the last place Denote: πtk0

0 = τ.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example : π = 5−2211−14131 π = 5−2211−14131 t0 → 305−2211−142 t0 → 41305−22110

t0

→ 1−141305−222 t0 → 211−141305−1 = τ so τ = πt4

0 = 211−14135−1.

In general, π = τtn−k0 , k0 = |π|−1(n).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

After we have discovered k0, we work solely with τ. Start with the identity element and multiply from the right by successive generators to get τ(= πt0

n−k0).

In the first step, multiply the identity from the right by tn−1 repeatedly kn−1 times until tkn−1

n−1 (n − 1) = τ(n − 1).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

After we have discovered k0, we work solely with τ. Start with the identity element and multiply from the right by successive generators to get τ(= πt0

n−k0).

In the first step, multiply the identity from the right by tn−1 repeatedly kn−1 times until tkn−1

n−1 (n − 1) = τ(n − 1).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

After we have discovered k0, we work solely with τ. Start with the identity element and multiply from the right by successive generators to get τ(= πt0

n−k0).

In the first step, multiply the identity from the right by tn−1 repeatedly kn−1 times until tkn−1

n−1 (n − 1) = τ(n − 1).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Continue in this way in order to place the other digits. Explicitly, multiply tkn−1

n−1 · · · tki i

by tki−1

i−1 in such a way that

tkn−1

n−1 · · · tki i tki−1 i−1 (i − 1) = τ(i − 1).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example, Cont.

Recall: π = 5−2211−14131, τ = 211−141305−1. 1020304050 t4 → 4−110203051 t−1

3

→ 1020403050 t−1

3

→ 204011305−1 t−1

3

→ 401121305−2 t−1

3

→ 112141305−3 t2 → 201141305−2 t2 → 102041305−1 t2 → 2−11041305

t2

→ 1−12−1413051 t2 → 2−21−1413052 t−3

1

→ 211−141305−1 = τ. Hence π = τt4

0 = t1 4t−4 3 t5 2t−3 1 t4 0.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Positive elements

Define: ˜ S+

n = {π = tkn−1 n−1 · · · tk1 1 tk0 0 | ki ≥ 0, 1 ≤ i ≤ n − 1}.

Theorem π = τtk0

0 ∈ ˜

S+

n−1 if and only if τ = [τ1, . . . , τn] (in integer

notation) satisfies: τi < 0 for 1 ≤ i ≤ n − 1. τn > 0. τi > τi−1 for 1 < i ≤ n. Example τ = [−6, −4, −3, −2, 30]

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

equidistribition with regard to positive elements

Recall: α(π) =

n−1

• i=0

ki. Theorem

• π∈˜

S+

n

qα(π) = [n]q (1 − q)n−1 =

• π∈˜

Sn

qℓ(π).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Equidistribution with regard to the whole group

The "Hilbert’s Hotel bijection" h : Z → N0 is defined by: h(x) =

• 2x

x ≥ 0 −2x − 1 x < 0 Use the ’Hilbert’s Hotel’s bijection h to build a bijection Φ : ˜ Sn → ˜ S+

n

by π′ = Φ(π) = Φ(tkn−1

n−1 · · · tk0 0 ) = th(kn−1) n−1

· · · th(k1)

1

tk0

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

The parameter ogsum

Let π ∈ ˜ Sn. Define

• gsum(π) = α(π′)

Theorem

• π∈˜

Sn

qogsum(π) = [n]q (1 − q)n−1 =

• π∈˜

Sn

qℓ(π).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Lehmer code of a permutation

The (shifted) Lehmer code of σ ∈ Sn is defined as a function Lσ : [n] → [n] by Lσ(i) = |{j | j < i, σ(j) < σ(i)}| + 1. Example: σ = 1 2 3 4 5 2 1 4 3 5

• (Lσ(1), Lσ(2), Lσ(3), Lσ(4), Lσ(5)) = (1, 1, 3, 3, 5).

For τ ∈ ˜ Sn, identify Lτ(i) with L|τ|(i).

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

A formula for the exponents of the O.G.S. generators

Let π ∈ ˜ Sn. Write π = τtk0

0 .

Denote for each 1 ≤ i ≤ n − 1: di = ci(τ), dn = 0. Theorem Let π = tkn−1

n−1 · · · tk1 1 tk0 0 = τ · tk0 0 .

Then for each 1 ≤ i ≤ n − 1: ki = Lτ(i + 1) + i · di+1 − (Lτ(i) + idi) − 1.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Define: negs(π) = |{i | Li+1(τ) − Li(τ) + i(di+1 − di) > 1}| Theorem

• gsum(π) = 2

n−1

• i=1

|Li+1(τ) − Li(τ) + i(di+1 − di) − 1|+|π|−1(n)−negs(π)

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example: π = 5−2211−14131 τ = 211−141305−1 (Lτ(1), Lτ(2), Lτ(3), Lτ(4), Lτ(5)) = (1, 1, 3, 3, 5).

(d1, d2, d3, d4, d5) = (1, −1, 1, 0, 0)

Lτ(2) − Lτ(1) + 1 · (d2 − d1) − 1 = −3 Lτ(3) − Lτ(2) + 2 · (d3 − d2) − 1 = 5 Lτ(4) − Lτ(3) + 3 · (d4 − d3) − 1 = −4 Lτ(5) − Lτ(4) + 4 · (d5 − d4) − 1 = 1 negs(π) = 2 |π|−1(5) = 1

• gsum

2 3 5 4 1 1 2 29

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

An algebraic view

˜ Sn acts on the polynomial ring C[x0, . . . , xn−1] by: si(xi−1) = xi si(xi) = xi−1 si(xj) = xj if j / ∈ {i − 1, i}. s0(x0) = 2xn−1. s0(xn−1) = 1

2x0

s0(xi) = xi for i / ∈ {0, n − 1}.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

The ideal of all elements fixed by ˜ Sn is generated by the monomial x0 · · · xn−1. The Coinvariant algebra is the quotient: R = C[x0, . . . , xn−1]/ < x0 · · · xn−1 >

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

The coinvariant algebra can be decomposed into homogenous parts in the following way: R =

• n≥0

Rn where Rn =

• a0···an−1=0

a0+···+an−1=n

C[xa0

0 · · · xan−1 n−1 ].

R has Hilbert series:

• n≥0

dimRnqn = [n]q (1 − q)n−1 .

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

A monomial basis for R, in bijection with the group elements

Define Bk = {xa0

0 · · · xan−1 n−1 | a0 · · · an−1 = 0, a0 · · · an−1 = 0}.

Bk is a monomial basis of Rk. Define the bijection Ψ : Bk → {π ∈ ˜ S+

n | ogsum(π) = k}

in the following way:

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Let m = xa0

0 xa1 1 · · · xan−1 n−1 ∈ B.

Write a0 = (n − 1) · q + b0 where b0 ∈ {1, . . . n − 1}. Define Ψ(m) = tbn−1

n−1 tbn−2 n−2 · · · tb0

where for i ∈ {1, . . . , n − 1}, bi = ai + q.

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Compose this map with the map Φ−1 : ˜ S+

n → ˜

Sn defined earlier in order to match a permutation with each basis element .

permutation statistics in the affine Coxeter group of type A

The Coxeter affine group of type A An algorithm for the O.G.S. decomposition Positive elements Equidistribution with length The parameter ogsum - A combinatorial view An algebraic view

Example

Let n = 5 and let m = x13

0 x2 1x1 2x0 3x1 4.

Then a0 = 13 = 4 · 3 + 1 b0 = 1 b1 = 2 + 3 = 5 b2 = 1 + 3 = 4 b3 = 0 + 3 = 3 b4 = 1 + 3 = 4 x13

0 x2 1x2x4 → t4 4t3 3t4 2t5 1t1 0 → t2 4t−1 3 t2 2t−3 1 t1 0.

permutation statistics in the affine Coxeter group of type A