Mahonians and parabolic quotients

Mahonians and parabolic quotients

FABRIZIO CASELLI

September 21, 2011

Poincar´ e polynomial

The Poincar´e polynomial of the symmetric groupS_n X

σ∈Sn

q^`(σ)

= [2]_q[3]_q· · ·[n]_q,

where [r]_q= 1 +q+. . .+q^r−1 and

`(σ) ={(i,j) :i <j, σ(i)> σ(j)} For a finite reflection groupW

X

u∈W

q^`(u)= [d1]q[d2]q· · ·[dr]q,

Poincar´ e polynomial

The Poincar´e polynomial of the symmetric groupS_n X

σ∈Sn

q^`(σ)= [2]_q[3]_q· · ·[n]_q,

where [r]_q= 1 +q+. . .+q^r−1 and

`(σ) ={(i,j) :i <j, σ(i)> σ(j)}

For a finite reflection groupW X

u∈W

q^`(u)= [d1]q[d2]q· · ·[dr]q,

Poincar´ e polynomial

The Poincar´e polynomial of the symmetric groupS_n X

σ∈Sn

q^`(σ)= [2]_q[3]_q· · ·[n]_q,

where [r]_q= 1 +q+. . .+q^r−1 and

`(σ) ={(i,j) :i <j, σ(i)> σ(j)}

For a finite reflection groupW X

u∈W

q^`(u)= [d1]q[d2]q· · ·[dr]q,

Parabolic subgroups and quotients

LetW =S_n ands_i = (i,i + 1) andS ={s₁, . . . ,sn−1}.

IfJ⊆S then

W_J is the subgroup genereted byJ;

JW is a system of coset representatives;

Ifσ ∈W there are uniqueσ_J ∈W_J and^Jσ∈^JW: σ =σ_J ·^Jσ and `(σ) =`(σ_J) +`(^Jσ).

X

Jσ∈^JW

q^`(Jσ)= P

σ∈W q^`(σ) P

σJ∈W_J q^`(σJ)

If J={s_n−k+1, . . . ,sn−1} then WJ ∼=Sk

X

σ∈^JW

q^`(σ)= [2]_q[3]_q· · ·[n]_q

[2]_q[3]_q· · ·[k]_q = [k+ 1]_q[k+ 2]_q· · ·[n]_q.

Parabolic subgroups and quotients

LetW =S_n ands_i = (i,i + 1) andS ={s₁, . . . ,sn−1}.

IfJ⊆S then

W_J is the subgroup genereted byJ;

JW is a system of coset representatives;

Ifσ ∈W there are uniqueσ_J ∈W_J and^Jσ∈^JW: σ =σ_J ·^Jσ and `(σ) =`(σ_J) +`(^Jσ).

X

Jσ∈^JW

q^`(Jσ)= P

σ∈W q^`(σ) P

σJ∈W_J q^`(σJ)

If J={s_n−k+1, . . . ,sn−1} then WJ ∼=Sk

X

σ∈^JW

q^`(σ)= [2]_q[3]_q· · ·[n]_q

[2]_q[3]_q· · ·[k]_q = [k+ 1]_q[k+ 2]_q· · ·[n]_q.

Parabolic subgroups and quotients

LetW =S_n ands_i = (i,i + 1) andS ={s₁, . . . ,sn−1}.

IfJ⊆S then

W_J is the subgroup genereted byJ;

JW is a system of coset representatives;

Ifσ ∈W there are uniqueσ_J ∈W_J and^Jσ∈^JW: σ =σ_J ·^Jσ and `(σ) =`(σ_J) +`(^Jσ).

X

Jσ∈^JW

q^`(Jσ)= P

σ∈W q^`(σ) P

σJ∈W_J q^`(σJ)

If J={s_n−k+1, . . . ,sn−1} then WJ ∼=Sk

X

σ∈^JW

q^`(σ)= [2]_q[3]_q· · ·[n]_q

[2]_q[3]_q· · ·[k]_q = [k+ 1]_q[k+ 2]_q· · ·[n]_q.

Parabolic subgroups and quotients

LetW =S_n ands_i = (i,i + 1) andS ={s₁, . . . ,sn−1}.

IfJ⊆S then

W_J is the subgroup genereted byJ;

JW is a system of coset representatives;

Ifσ ∈W there are uniqueσ_J ∈W_J and^Jσ∈^JW: σ =σ_J ·^Jσ and `(σ) =`(σ_J) +`(^Jσ).

X

Jσ∈^JW

q^`(Jσ)= P

σ∈W q^`(σ) P

σJ∈W_J q^`(σJ)

If J={s_n−k+1, . . . ,sn−1} then WJ ∼=Sk

X

σ∈^JW

q^`(σ)= [2]_q[3]_q· · ·[n]_q

[2]_q[3]_q· · ·[k]_q = [k+ 1]_q[k+ 2]_q· · ·[n]_q.

Parabolic subgroups and quotients

LetW =S_n ands_i = (i,i + 1) andS ={s₁, . . . ,sn−1}.

IfJ⊆S then

W_J is the subgroup genereted byJ;

JW is a system of coset representatives;

Ifσ ∈W there are uniqueσ_J ∈W_J and^Jσ∈^JW: σ =σ_J ·^Jσ and `(σ) =`(σ_J) +`(^Jσ).

X

Jσ∈^JW

q^`(Jσ)= P

σ∈W q^`(σ) P

σJ∈W_J q^`(σJ)

If J={s_n−k+1, . . . ,sn−1} then W_J ∼=S_k X q^`(σ)= [2]_q[3]_q· · ·[n]_q

= [k+ 1] [k+ 2] · · ·[n] .

The major index in parabolic quotients

We let

Descents ofσ = Des(σ) ={i|σ(i)> σ(i+ 1)}

Major index of σ = maj(σ) = X

i∈Des(σ)

i

We have X

σ∈Sn

q^`(σ) = X

σ∈Sn

q^maj(σ) BUTmaj(σ)6=maj(σ_J) +maj(^Jσ).

Nevertheless,

Theorem (Panova, 2010)

If W =Sn and J={s_n−k+1, . . . ,sn−1} then X

σ∈^JW

q^maj(σ) = [k+ 1]q[k+ 2]q· · ·[n]q

The major index in parabolic quotients

We let

Descents ofσ = Des(σ) ={i|σ(i)> σ(i+ 1)}

Major index of σ = maj(σ) = X

i∈Des(σ)

i

We have X

σ∈Sn

q^`(σ) = X

σ∈Sn

q^maj(σ) BUTmaj(σ)6=maj(σ_J) +maj(^Jσ).

Nevertheless,

Theorem (Panova, 2010)

If W =Sn and J={s_n−k+1, . . . ,sn−1} then X

σ∈^JW

q^maj(σ) = [k+ 1]q[k+ 2]q· · ·[n]q

The major index in parabolic quotients

We let

Descents ofσ = Des(σ) ={i|σ(i)> σ(i+ 1)}

Major index of σ = maj(σ) = X

i∈Des(σ)

i

We have X

σ∈Sn

q^`(σ)= X

σ∈Sn

q^maj(σ)

BUTmaj(σ)6=maj(σ_J) +maj(^Jσ).

Nevertheless,

Theorem (Panova, 2010)

If W =Sn and J={s_n−k+1, . . . ,sn−1} then X

σ∈^JW

q^maj(σ) = [k+ 1]q[k+ 2]q· · ·[n]q

The major index in parabolic quotients

We let

Descents ofσ = Des(σ) ={i|σ(i)> σ(i+ 1)}

Major index of σ = maj(σ) = X

i∈Des(σ)

i

We have X

σ∈Sn

q^`(σ)= X

σ∈Sn

q^maj(σ) BUTmaj(σ)6=maj(σJ) +maj(^Jσ).

Nevertheless,

Theorem (Panova, 2010)

If W =Sn and J={s_n−k+1, . . . ,sn−1} then X

σ∈^JW

q^maj(σ) = [k+ 1]q[k+ 2]q· · ·[n]q

The major index in parabolic quotients

We let

Descents ofσ = Des(σ) ={i|σ(i)> σ(i+ 1)}

Major index of σ = maj(σ) = X

i∈Des(σ)

i

We have X

σ∈Sn

q^`(σ)= X

σ∈Sn

q^maj(σ) BUTmaj(σ)6=maj(σJ) +maj(^Jσ).

Nevertheless,

Theorem (Panova, 2010)

If W =Snand J ={s_n−k+1, . . . ,sn−1} then X

σ∈^JW

q^maj(σ) = [k+ 1]q[k+ 2]q· · ·[n]q

Signed Mahonians

An alternating version of the Poincar´e polynomial Theorem (Gessel-Simion)

X

σ∈S_n

(−1)^`(σ)q^maj(σ)

= [2]−q[3]q[4]−q· · ·[n]₍₋₁₎ⁿ⁻¹_q.

Problem: for J ={s_n−k+1, . . . ,sn−1} compute the polynomial X

σ∈^JW

(−1)^`(σ)q^maj(σ).

Does it factorize nicely? Is it an alternating version of Panova’s result? Yes. Yes.

Signed Mahonians

An alternating version of the Poincar´e polynomial Theorem (Gessel-Simion)

X

σ∈S_n

(−1)^`(σ)q^maj(σ) = [2]−q[3]q[4]−q· · ·[n]₍₋₁₎ⁿ⁻¹_q.

Problem: for J ={s_n−k+1, . . . ,sn−1} compute the polynomial X

σ∈^JW

(−1)^`(σ)q^maj(σ).

Does it factorize nicely? Is it an alternating version of Panova’s result? Yes. Yes.

Signed Mahonians

An alternating version of the Poincar´e polynomial Theorem (Gessel-Simion)

X

σ∈S_n

(−1)^`(σ)q^maj(σ) = [2]−q[3]q[4]−q· · ·[n]₍₋₁₎ⁿ⁻¹_q.

Problem: for J ={s_n−k+1, . . . ,sn−1} compute the polynomial X

σ∈^JW

(−1)^`(σ)q^maj(σ).

Does it factorize nicely?

Is it an alternating version of Panova’s result? Yes. Yes.

Signed Mahonians

An alternating version of the Poincar´e polynomial Theorem (Gessel-Simion)

X

σ∈S_n

(−1)^`(σ)q^maj(σ) = [2]−q[3]q[4]−q· · ·[n]₍₋₁₎ⁿ⁻¹_q.

Problem: for J ={s_n−k+1, . . . ,sn−1} compute the polynomial X

σ∈^JW

(−1)^`(σ)q^maj(σ).

Does it factorize nicely? Is it an alternating version of Panova’s result?

Yes. Yes.

Signed Mahonians

An alternating version of the Poincar´e polynomial Theorem (Gessel-Simion)

X

σ∈S_n

(−1)^`(σ)q^maj(σ) = [2]−q[3]q[4]−q· · ·[n]₍₋₁₎ⁿ⁻¹_q.

Problem: for J ={s_n−k+1, . . . ,sn−1} compute the polynomial X

σ∈^JW

(−1)^`(σ)q^maj(σ).

Does it factorize nicely? Is it an alternating version of Panova’s result? Yes. Yes.

An idea of Adin-Gessel-Roichman

Difficult to generalize Panova’s and Wachs’s proofs.

Use a catalytic parameter! (after an idea of Adin-Gessel-Roichman)

JW ={σ= [. . . ,n−k+ 1, . . . ,n−k+ 2, . . . ,n, . . .]}. Example

Ifn= 5 andk = 3 then

JW ={[12345],[13452],[21345],[23145],[23451],[31245], . . .} Let

s(σ) :=

σ(n)−1, ifσ(n)∈[n−k]; n−k, otherwise.

An idea of Adin-Gessel-Roichman

Difficult to generalize Panova’s and Wachs’s proofs.

Use a catalytic parameter!

(after an idea of Adin-Gessel-Roichman)

JW ={σ= [. . . ,n−k+ 1, . . . ,n−k+ 2, . . . ,n, . . .]}. Example

Ifn= 5 andk = 3 then

JW ={[12345],[13452],[21345],[23145],[23451],[31245], . . .} Let

s(σ) :=

σ(n)−1, ifσ(n)∈[n−k]; n−k, otherwise.

An idea of Adin-Gessel-Roichman

Difficult to generalize Panova’s and Wachs’s proofs.

Use a catalytic parameter! (after an idea of Adin-Gessel-Roichman)

JW ={σ= [. . . ,n−k+ 1, . . . ,n−k+ 2, . . . ,n, . . .]}. Example

Ifn= 5 andk = 3 then

JW ={[12345],[13452],[21345],[23145],[23451],[31245], . . .} Let

s(σ) :=

σ(n)−1, ifσ(n)∈[n−k]; n−k, otherwise.

An idea of Adin-Gessel-Roichman

Difficult to generalize Panova’s and Wachs’s proofs.

Use a catalytic parameter! (after an idea of Adin-Gessel-Roichman)

JW ={σ= [. . . ,n−k+ 1, . . . ,n−k+ 2, . . . ,n, . . .]}.

Example

Ifn= 5 andk = 3 then

JW ={[12345],[13452],[21345],[23145],[23451],[31245], . . .}

Let

s(σ) :=

σ(n)−1, ifσ(n)∈[n−k]; n−k, otherwise.

An idea of Adin-Gessel-Roichman

Difficult to generalize Panova’s and Wachs’s proofs.

Use a catalytic parameter! (after an idea of Adin-Gessel-Roichman)

JW ={σ= [. . . ,n−k+ 1, . . . ,n−k+ 2, . . . ,n, . . .]}.

Example

Ifn= 5 andk = 3 then

JW ={[12345],[13452],[21345],[23145],[23451],[31245], . . .}

Let

s(σ) :=

σ(n)−1, ifσ(n)∈[n−k];

n−k, otherwise.

A recursion

We let

f_n,k(q,z) = X

σ∈^JW

^`(σ)q^maj(σ)z^s(σ)

where=−1.

Theorem

For k= 1,2, . . . ,n−1

f_n,k(q,z) = 1 1 +z

^kz^n−k + (−q)ⁿ⁻¹

fn−1,k(q,1) + +ⁿz(1−qⁿ⁻¹)fn−1,k(q,−z)

+z^n−kfn−1,k−1(q,1). Does not restrict to a recursion forfn,k(q,1).

A recursion

We let

f_n,k(q,z) = X

σ∈^JW

^`(σ)q^maj(σ)z^s(σ)

where=−1.

Theorem

For k= 1,2, . . . ,n−1

f_n,k(q,z) = 1 1 +z

^kz^n−k + (−q)ⁿ⁻¹

fn−1,k(q,1) + +ⁿz(1−qⁿ⁻¹)fn−1,k(q,−z)

+z^n−kfn−1,k−1(q,1).

Does not restrict to a recursion forfn,k(q,1).

Explicit formulas

Now guess a formula and prove it.

Theorem (C, 2011) If k<n is odd we have

fn,k(q,z) = [k+ 1]−q[k+ 2]q· · ·[n−1]ⁿq

·^n−k−1X

i=0

(n+1)(n−i−1)zⁱqⁿ⁻ⁱ⁻¹+z^n−k[k]ⁿ⁻¹_q .

If k<n−1 is even we have

fn,k(q,z) = [k+ 2]_−q· · ·[n−1]ⁿq·

[k+ 1]ⁿq[n]n−1q+ (z −1)

·^n−k−1X

i=0

[k+ 1]ⁿq[n−i−1]n+1qzⁱ+

n−k−1

X

i=0 ieven

qⁿ⁻ⁱ⁻¹zⁱ [k]_−q−[k]q .

Explicit formulas

Now guess a formula and prove it.

Theorem (C, 2011) If k<n is odd we have

fn,k(q,z) = [k+ 1]−q[k+ 2]q· · ·[n−1]ⁿq

·^n−k−1X

i=0

(n+1)(n−i−1)zⁱqⁿ⁻ⁱ⁻¹+z^n−k[k]ⁿ⁻¹_q .

If k<n−1 is even we have

fn,k(q,z) = [k+ 2]_−q· · ·[n−1]ⁿq·

[k+ 1]ⁿq[n]n−1q+ (z −1)

·^n−k−1X

i=0

[k+ 1]ⁿq[n−i−1]n+1qzⁱ+

n−k−1

X

i=0 ieven

qⁿ⁻ⁱ⁻¹zⁱ [k]_−q−[k]q .

Explicit formulas

Now guess a formula and prove it.

Theorem (C, 2011) If k<n is odd we have

fn,k(q,z) = [k+ 1]−q[k+ 2]q· · ·[n−1]ⁿq

·^n−k−1X

i=0

(n+1)(n−i−1)zⁱqⁿ⁻ⁱ⁻¹+z^n−k[k]ⁿ⁻¹_q .

If k<n−1 is even we have

fn,k(q,z) = [k+ 2]_−q· · ·[n−1]ⁿq·

[k+ 1]ⁿq[n]n−1q+ (z−1)

·^n−k−1X

[k+ 1]ⁿq[n−i−1]n+1qzⁱ+

n−k−1

X qⁿ⁻ⁱ⁻¹zⁱ [k]_−q−[k]q .

The specialization

Corollary

For J ={s_n−k+1,sn−k+2, . . . ,sn−1} we have f_n,k(q,1) = X

σ∈^JW

^`(σ)q^maj(σ)

= [k+ 1]k+n+nkq[k+ 2]k+1q[k+ 3]k+2q· · ·[n]ⁿ⁻¹_q.

Hope someone will be able to explain this result. I have only been able to prove it.

The specialization

Corollary

For J ={s_n−k+1,sn−k+2, . . . ,sn−1} we have f_n,k(q,1) = X

σ∈^JW

^`(σ)q^maj(σ)

= [k+ 1]k+n+nkq[k+ 2]k+1q[k+ 3]k+2q· · ·[n]ⁿ⁻¹_q. Hope someone will be able to explain this result.

I have only been able to prove it.

The specialization

Corollary

For J ={s_n−k+1,sn−k+2, . . . ,sn−1} we have f_n,k(q,1) = X

σ∈^JW

^`(σ)q^maj(σ)

= [k+ 1]k+n+nkq[k+ 2]k+1q[k+ 3]k+2q· · ·[n]ⁿ⁻¹_q. Hope someone will be able to explain this result.

I have only been able to prove it.

Complex reflection groups

The group of r-colored permutations:

G(r,n) ={[σ₁^z1, . . . , σ_n^zn] : σ∈S_n and z_i ∈Zr}.

The infinite family of irreducible complex reflection groups: if p|r,

G =G(r,p,n) ={[σ^z₁¹, . . . , σ_n^zn]∈G(r,n) : z₁+· · ·+z_n≡0 mod p}.

And other related groups: we let Cp=h[1^r/p, . . . ,n^r/p]i and G^∗ :=G(r,n)/C_p.

Complex reflection groups

The group of r-colored permutations:

G(r,n) ={[σ₁^z1, . . . , σ_n^zn] : σ∈S_n and z_i ∈Zr}.

The infinite family of irreducible complex reflection groups: if p|r,

G =G(r,p,n) ={[σ^z₁¹, . . . , σ_n^zn]∈G(r,n) : z₁+· · ·+z_n≡0 mod p}.

And other related groups: we let Cp=h[1^r/p, . . . ,n^r/p]i and G^∗ :=G(r,n)/C_p.

Complex reflection groups

The group of r-colored permutations:

G(r,n) ={[σ₁^z1, . . . , σ_n^zn] : σ∈S_n and z_i ∈Zr}.

The infinite family of irreducible complex reflection groups: if p|r,

G =G(r,p,n) ={[σ^z₁¹, . . . , σ_n^zn]∈G(r,n) : z₁+· · ·+z_n≡0 mod p}.

And other related groups: we let Cp=h[1^r/p, . . . ,n^r/p]i and G^∗ :=G(r,n)/C_p.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then

g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴]. The exponents are

non-increasing;

strict at the “homogeneous” descents; as small as possible with these properties. We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n). X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents; as small as possible with these properties. We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n). X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents;

as small as possible with these properties. We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n). X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents;

as small as possible with these properties.

We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n). X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents;

as small as possible with these properties.

We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n). X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents;

as small as possible with these properties.

We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n).

X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

Flag-major index

Ifg = [2³ ,5¹ ,4⁰ ,7⁵ ,3⁵,1⁴,6⁴]∈G(6,7) then g = [2¹⁵,5¹³,4¹²,7¹¹,3⁵,1⁴,6⁴].

The exponents are non-increasing;

strict at the “homogeneous” descents;

as small as possible with these properties.

We letλ(g) = (15,13,12,11,5,4,4) and fmaj(g) =|λ(g)|= 15 + 13 +· · ·+ 4 = 64.

Originally defined by Adin and Roichman for the groupG(r,n).

X

g∈G^∗

q^fmaj(g)= [d₁]_q[d₂]_q· · ·[d_n]_q,

wheredi are the fundamental degrees ofG.

A bijection ` a la Garsia-Gessel

Want to extend Panova’s result to these groups.

Lemma The map

G^∗× P_n× {0,1, . . . ,p−1} −→ Nⁿ

(g, λ,h) 7→ f = (f1, . . . ,fn), where fi =λ_|g⁻¹_(i)|(g) +rλ_|g⁻¹_(i)|+h^r_p for all i ∈[n], is a bijection. And in this case we say that f is g -compatible. Fork <n we let

C_k ={[σ₁⁰, σ₂⁰, . . . , σ⁰_k,g_k+1, . . . ,g_n]∈G^∗ : σ₁<· · ·< σ_k}.

A bijection ` a la Garsia-Gessel

Want to extend Panova’s result to these groups.

Lemma The map

G^∗× P_n× {0,1, . . . ,p−1} −→ Nⁿ

(g, λ,h) 7→ f = (f₁, . . . ,f_n), where fi =λ_|g⁻¹_(i)|(g) +rλ_|g⁻¹_(i)|+h^r_p for all i ∈[n], is a bijection. And in this case we say that f is g -compatible.

Fork <n we let

C_k ={[σ₁⁰, σ₂⁰, . . . , σ⁰_k,g_k+1, . . . ,g_n]∈G^∗ : σ₁<· · ·< σ_k}.

A bijection ` a la Garsia-Gessel

Want to extend Panova’s result to these groups.

Lemma The map

G^∗× P_n× {0,1, . . . ,p−1} −→ Nⁿ

(g, λ,h) 7→ f = (f₁, . . . ,f_n), where fi =λ_|g⁻¹_(i)|(g) +rλ_|g⁻¹_(i)|+h^r_p for all i ∈[n], is a bijection. And in this case we say that f is g -compatible.

Fork <n we let

C_k ={[σ₁⁰, σ₂⁰, . . . , σ⁰_k,g_k+1, . . . ,g_n]∈G^∗ : σ₁<· · ·< σ_k}.

The result

Theorem (C. 2011) Let G =G(r,p,n)^∗. Then

X

g∈C_k

q^fmaj(g−1)= [p]_qkr/p[r(k+ 1)]_q· · ·[r(n−1)]_q[rn/p]_q.

Corollary

If G =G(r,n), then Ck is a system of coset representatives for the (parabolic) subgroup G(r,k) and

X

g∈C_k

q^fmaj(g−1) = [r(k+ 1)]_q[r(k+ 2)]_q· · ·[rn]_q.

The result

Theorem (C. 2011) Let G =G(r,p,n)^∗. Then

X

g∈C_k

q^fmaj(g−1)= [p]_qkr/p[r(k+ 1)]_q· · ·[r(n−1)]_q[rn/p]_q.

Corollary

If G =G(r,n), then Ck is a system of coset representatives for the (parabolic) subgroup G(r,k) and

X

g∈C_k

q^fmaj(g−1) = [r(k+ 1)]_q[r(k+ 2)]_q· · ·[rn]_q.

Longest increasing subsequence

Elements starting with a longest 0-colored increasing subsequence Π_r,n,k := {g = [σ⁰₁, . . . , σ_n−k⁰ , σ_n−k+1^zn−k+1, . . . , σ^z_nⁿ]∈G(r,n) :

σ₁ <· · ·< σn−k and no increasing subsequence of lengthn−k+ 1colored with 0}.

Theorem

If n≥2k we have that X

g∈Π_r,n,k

q^fmaj(g−1)=

k

X

i=0

(−1)ⁱ k

i

[r(n−i+1)]_q[r(n−i+2)]_q· · ·[rn]_q.

Longest increasing subsequence

Elements starting with a longest 0-colored increasing subsequence Π_r,n,k := {g = [σ⁰₁, . . . , σ_n−k⁰ , σ_n−k+1^zn−k+1, . . . , σ^z_nⁿ]∈G(r,n) :

σ₁ <· · ·< σn−k and no increasing subsequence of lengthn−k+ 1colored with 0}.

Theorem

If n≥2k we have that X

g∈Π_r,n,k

q^fmaj(g−1)=

k

X

i=0

(−1)ⁱ k

i

[r(n−i+1)]_q[r(n−i+2)]q· · ·[rn]q.

Open problems

Problem

Let J⁰ = [k]. Numerical evidence shows that X

σ∈^J0Sn

(−1)^`(σ)q^maj(σ)= X

u∈^JSn

(−1)^`(σ)q^maj(σ)

if and only if n is odd or k is even (or both). Give a (possibly bijective) proof of this phenomenon.

Problem

Unify the main results of this work in a unique statement, i.e. compute the polynomials

X

g∈Ck

^`(|g|)q^fmaj(g−1).

This is known to have nice factorization ifk = 0 (Biagioli-C.)

Open problems

Problem

Let J⁰ = [k]. Numerical evidence shows that X

σ∈^J0Sn

(−1)^`(σ)q^maj(σ)= X

u∈^JSn

(−1)^`(σ)q^maj(σ)

if and only if n is odd or k is even (or both). Give a (possibly bijective) proof of this phenomenon.

Problem

Unify the main results of this work in a unique statement, i.e.

compute the polynomials X

g∈C

^`(|g|)q^fmaj(g−1).

This is known to have nice factorization ifk = 0 (Biagioli-C.)

Open problems

Problem

Let J⁰ = [k]. Numerical evidence shows that X

σ∈^J0Sn

(−1)^`(σ)q^maj(σ)= X

u∈^JSn

(−1)^`(σ)q^maj(σ)

if and only if n is odd or k is even (or both). Give a (possibly bijective) proof of this phenomenon.

Problem

Unify the main results of this work in a unique statement, i.e.

compute the polynomials X

g∈C_k

^`(|g|)q^fmaj(g−1).

This is known to have nice factorization ifk = 0 (Biagioli-C.)