### A length function for the complex reflection group G (r , r , n)

Eli Bagno and Mordechai Novick

SLC 78, March 28, 2017

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### General Definitions

S_{n} is the symmetric group on {1, . . . ,n}.

Zr is the cyclic group of orderr. ζr is the primitiver−th root of unity.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Complex reflection groups

G(r,n) = group of all matrices π= (σ,k), where:

σ =a1· · ·an∈Sn.

k = (k1, . . . ,kn)∈Z^{n}_{r}.(k-vector)

π = (σ,k) is the n×n monomial matrix with non-zero entries
ζ_{r}^{k}^{i} in the (ai,i) positions.

Example (n= 3,r = 4)

π(312,(1,3,3)) =

0 i 0

0 0 −i

−i 0 0

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

For p|r,G(r,p,n) is the subgroup of G(r,n) consisting of matrices (σ,k) satisfying

n

Y

i=1

(ζ_{r}^{k}^{i})^{r}^{p} = 1.

Hence G(r,r,n) is the group of such matrices satisfying:

n

Y

i=1

(ζ_{r}^{k}^{i}) = 1

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### One-line notation

We denote an element ofG(r,p,n) in a more concise manner:

(σ,k) =a^{k}_{1}^{1}· · ·a^{k}_{n}^{n}
forσ =a1· · ·an and k = (k1, . . . ,kn).

Example

π(312,(1,3,3)) = 3^{1}1^{3}2^{3}

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Our goal

Various sets of generators have been defined for complex reflection groups but (as far as we know), no length function has been formulated.

We provide such a function for the case ofG(r,r,n) with a specific choice of generating set proposed by Shi.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Shi’s Generators for G (r , r , n)

For eachi ∈ {1, . . . ,n−1} let si = (i,i+ 1) be the familiar
adjacent transpositions generatingS_{n}.

Definet0 = (1^{r−1},n^{1}).

Theorem

The set{t_{0},s_{1}, . . . ,sn−1} generates G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Example of generators acting from the right

Applyings1 from the right:

π= 3^{0}2^{2}1^{−1}4^{−1} 7→2^{2}3^{0}1^{−1}4^{−1}
Applyingt_{0} from the right:

π= 2^{0}1^{2}3^{−1}4^{−1} 7→4^{−2}1^{2}3^{−1}2^{1}
Remark

Places are exchanged, the k−vector is not preserved.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Example of generators acting from the left

Applyings1 from the left:

π= 2^{0}1^{2}3^{−1}4^{−1} 7→1^{0}2^{2}3^{−1}4^{−1}
Applyingt_{0} from the left:

π= 2^{0}1^{2}3^{−1}4^{−1} 7→2^{0}4^{2}3^{−1}1^{−1}
Remark

Numbers are exchanged and the k-vector is preserved.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### The affine group

The affine Weyl group ˜Sn is defined as follows:

S˜n={w :Z→Z|w(i+n) =w(i)+n,∀i∈ {1, . . . ,n},

n

X

i=1

w(i) = n+ 1

2

}.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

Each affine permutation can be written ininteger window notation in the form:

π = (π(1), . . . , π(n)) = (b1, . . . ,bn).

By writingbi =n·ki+ai, we can use theresidue window notation:

π =a^{k}_{1}^{1}· · ·a^{k}_{n}^{n}.

where{a_{1}, . . . ,an}={1, . . . ,n}.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Generators for the affine group

For eachi ∈ {1, . . . ,n−1} let si = (i,i+ 1) be the known adjacent transpositions generatingSn.

Defines_{0}= (1,n^{−1}).

generators.PNG generators.PNG

Figure:

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

Theorem

Letπ=a^{k}_{1}^{1}· · ·a^{k}_{n}^{n} ∈S˜n. Then

`(π) = X

1≤i<j≤n ai<aj

|k_{j} −k_{i}|+ X

1≤i<j≤n ai>aj

|k_{j} −k_{i} −1|

Example

Ifπ= 3^{−1}1^{0}4^{1}2^{0} then:

`(π) =|1−(−1)|+|1−0|+|0−(−1)−1|+|0−(−1)−1|+|0−1−1|= 5

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Another presentation of ˜ S

_{n}

Each affine permutationπ=a^{k}_{1}^{1}· · ·a^{k}_{n}^{n} can also be written as a
monomial matrix:

Mπ = (mij) =

(0 i 6=σ(j)
x^{k}^{i} i =σ(j)

Example (n= 4)

π = 3^{−1}1^{0}4^{1}2^{0} =

0 x^{0} 0 0

0 0 0 x^{0}

x^{−1} 0 0 0

0 0 x^{1} 0

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Mapping ˜ S

_{n}

### to G (r , r , n)

Shi defines a homomorphism η: ˜S_{n}→G(r,r,n) by
substituting a primitive r-th root of unityζ_{r} in place of x.

He tried to adapt his length function for the affine groups to the case of G(r,r,n) but did not obtain a closed formula.

Here we provide such a formula.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Difficulties in adapting Shi’s formula

InG(r,r,n) each element does not have a uniquely defined k-
vector, as adding a multiple ofr to anyk_{i} does not change π as an
element ofG(r,r,n).

Example

The permutations 4^{5}2^{−4}3^{−2}1^{1} and 4^{0}2^{−4}3^{3}1^{1} represent the same
element ofG(5,5,4).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### The normal form

Definition

A permutation(p,k^{0})∈G(r,r,n)is said to be in normal formif
the following conditions are met:

1

n

P

i=1

k_{i}^{0}= 0

2 |max(k^{0})−min(k^{0})| ≤r

3 If there exist i <j such that |k_{j}^{0}−k_{i}^{0}|=r then k_{j}^{0}−k_{i}^{0}=r .
If(p,k^{0}) is in normal form and is equivalent to(p,k) then we say
that(p,k^{0}) is a normal form of(p,k).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

Example

The normal form of 4^{−8}1^{15}3^{12}2^{9}∈G(7,7,4) is 4^{−1}1^{1}3^{−2}2^{2}.
Theorem

For eachπ ∈G(r,r,n) a normal form exists and is unique.

Shi’s length function, when applied to all representatives of a permutation in G(r,r,n), attains its minimum on the normal form representative.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Decomposition Into Right Cosets of S

_{n}

Let π= (k, σ)∈G(r,r,n).

As we have seen, for each generator τ of Sn ,π andτ π have the same k-vector.

Hence, it is natural and straightforward to decompose G(r,r,n) into right cosets.

Each right coset has a unique representativeπ = (k, σ) which has minimal length.

This leads us to a new length function for G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### The length function for G (r , r , n)

Letπ=a^{k}_{1}^{1}· · ·a^{k}_{n}^{n} ∈G(r,r,n).

Writeπ=u·σ whereu ∈S_{n} andσ is the minimal length
representative. Then:

Theorem

`(π) = P

1≤i<j≤n

|k_{j}−k_{i}| −noninv(k) +inv(u)

where

noninv(k) = #{(i,j)|i <j,k(i)<k(j)}

and (as usual)

inv(u) = #{(i,j)|i <j,u(i)>u(j)}.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Length Example

Letπ= 3^{1}1^{−2}2^{0}4^{1}∈G(4,4,4).

Thenσ = 1^{1}4^{−2}3^{0}2^{1}, andu =|π||σ|^{−1} = 3421.

Hence:

X

1≤i<j≤n

|k_{j}−k_{i}|=|−2−1|+|0−1|+|1−1|+|0−(−2)|+|1−(−2)|+|1−0|= 10

And:

noninv(k) = 3 while

inv(u) = 5 so that`(π) = 10−3 + 5 = 12

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Finding the minimal-length representative

The minimal-length elementσ=a^{k}_{1}^{1}· · ·a^{k}_{n}^{n} ∈G(r,r,n) for the
k-vector (k1, . . . ,kn)

(abbreviateda_{1}a_{2}· · ·a_{n}∈S_{n})

is the unique one with the following property:

ai <aj iff:

k(i)>k(j), or k(i) =k(j) andi <j Example

Ifk= (−2,1,−1,1,2,−1) thenσ = 624315

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### Open question: What is the generating function?

LetGr,r,n(q) = P

π∈G_{r,r,n}

q^{`(π)}.

From the coset decomposition it is clear thatGr,r,n(q) has [n]q! as a factor.

Example

G4,4,4(q) = [4]q!(1+2q^{2}+3q^{3}+4q^{4}+5q^{5}+7q^{6}+8q^{7}+10q^{8}+12q^{9}+7q^{10}+3q^{11})

G6,6,3(q) = [3]q!(1+q+2q^{2}+2q^{3}+3q^{4}+3q^{5}+4q^{6}+4q^{7}+5q^{8}+5q^{9}+6q^{10})

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### A possible direction...

There is a bijection between left cosets ofSn in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)).

In B-B, each partition is theinversion table of the

corresponding left coset (i.e., of its ascending minimal-length representative).

The bijection in E-E maps each left coset to the conjugate of its inversion table.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### A possible direction...

There is a bijection between left cosets ofSn in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)).

In B-B, each partition is theinversion table of the

corresponding left coset (i.e., of its ascending minimal-length representative).

The bijection in E-E maps each left coset to the conjugate of its inversion table.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

### A possible direction...

There is a bijection between left cosets ofSn in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)).

In B-B, each partition is theinversion table of the

corresponding left coset (i.e., of its ascending minimal-length representative).

The bijection in E-E maps each left coset to the conjugate of its inversion table.

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

This correspondence yields the following generating function for length in the affine group:

S˜_{n}(q) = [n]_{q}!

(1−q)(1−q^{2})· · ·(1−q^{n})

A similar approach may work in our case of right cosets in G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

This correspondence yields the following generating function for length in the affine group:

S˜_{n}(q) = [n]_{q}!

(1−q)(1−q^{2})· · ·(1−q^{n})

A similar approach may work in our case of right cosets in G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

This correspondence yields the following generating function for length in the affine group:

S˜_{n}(q) = [n]_{q}!

(1−q)(1−q^{2})· · ·(1−q^{n})

A similar approach may work in our case of right cosets in G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

This correspondence yields the following generating function for length in the affine group:

S˜_{n}(q) = [n]_{q}!

(1−q)(1−q^{2})· · ·(1−q^{n})

A similar approach may work in our case of right cosets in G(r,r,n).

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)

## Thank you!!

Eli Bagno and Mordechai Novick A length function for the complex reflection groupG(r,r,n)