Leibniz Universität Hannover

Multiline queues with spectral parameters

Darij Grinberg

joint work with Erik Aas and Travis Scrimshaw

28 June 2018

Leibniz Universit¨at Hannover

slides: http://www.cip.ifi.lmu.de/~grinberg/algebra/

hannover2018.pdf paper:

http://www.cip.ifi.lmu.de/~grinberg/algebra/mlqs.pdf

1 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n∈Z/nZas sites. Regard them as points on a line that “wraps around” cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}. Ifu is a word andi is a site, then u_i :=u(i). Writeu₁u₂· · ·u_nfor a word u (“one-line notation”). Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n ∈Z/nZas sites.

Regard them as points on a line that “wraps around”

cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}. Ifu is a word andi is a site, then u_i :=u(i). Writeu₁u₂· · ·u_nfor a word u (“one-line notation”). Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n ∈Z/nZas sites.

Regard them as points on a line that “wraps around”

cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}.

Ifu is a word andi is a site, then u_i :=u(i).

Writeu₁u₂· · ·u_nfor a word u (“one-line notation”). Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n ∈Z/nZas sites.

Regard them as points on a line that “wraps around”

cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}.

Ifu is a word andi is a site, then u_i :=u(i).

Writeu₁u₂· · ·u_n for a wordu (“one-line notation”).

Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n ∈Z/nZas sites.

Regard them as points on a line that “wraps around”

cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}.

Ifu is a word andi is a site, then u_i :=u(i).

Writeu₁u₂· · ·u_n for a wordu (“one-line notation”).

Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Sites and words

We study a combinatorial algorithm: the action of queues on words.

Fix a positive integern.

For a nonnegative integer k, let [k] be the set {1,2, . . . ,k}.

We shall refer to the elements 1,2, . . . ,n ∈Z/nZas sites.

Regard them as points on a line that “wraps around”

cyclically:

· · · n−1 n 1 2 · · · n−1 n 1 2 · · ·

A word means a map{sites} → {positive integers}.

Ifu is a word andi is a site, then u_i :=u(i).

Writeu₁u₂· · ·u_n for a wordu (“one-line notation”).

Example: The word 33122 (for n= 5) is the map

i = · · · 4 5 1 2 3 4 5 1 2 · · ·

7→ u_i = · · · 2 2 3 3 1 2 2 3 3 · · ·

2 / 23

Queues

A queuemeans a set of sites.

Draw a queueq by putting circles on all the sitesi ∈q.

Example: The queue {2,5} (forn= 7) is represented by We shall omit all the grey parts in the future (i.e., we will draw only one copy of each site).

3 / 23

Queues

A queuemeans a set of sites.

Draw a queueq by putting circles on all the sitesi ∈q.

Example: The queue {2,5} (forn= 7) is represented by

We shall omit all the grey parts in the future (i.e., we will draw only one copy of each site).

3 / 23

Queues

A queuemeans a set of sites.

Draw a queueq by putting circles on all the sitesi ∈q.

Example: The queue {2,5} (forn= 7) is represented by

· · · 6 7 1 2 3 4 5 6 7 1 2 · · ·

We shall omit all the grey parts in the future (i.e., we will draw only one copy of each site).

3 / 23

Queues

A queuemeans a set of sites.

Draw a queueq by putting circles on all the sitesi ∈q.

Example: The queue {2,5} (forn= 7) is represented by

· · · 6 7 1 2 3 4 5 6 7 1 2 · · ·

We shall omit all the grey parts in the future (i.e., we will draw only one copy of each site).

3 / 23

Queues

A queuemeans a set of sites.

Draw a queueq by putting circles on all the sitesi ∈q.

Example: The queue {2,5} (forn= 7) is represented by

1 2 3 4 5 6 7

We shall omit all the grey parts in the future (i.e., we will draw only one copy of each site).

3 / 23

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

4 / 23

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

4 / 23

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7

Phase I: For each of the largestn− |q|letters ofu (in decreasing order),

drop this letterdown andadd 1 to it;

moveit left until hitting some unoccupied sitei ∈/ q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7

Phase I: For each of the largestn− |q|letters ofu (in decreasing order),

drop this letterdown andadd 1 to it;

moveit left until hitting some unoccupied sitei ∈/ q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7 5

Phase I: For each of the largestn− |q|letters ofu (in decreasing order),

drop this letterdown andadd 1 to it;

moveit left until hitting some unoccupied sitei ∈/ q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7 4 5

Phase I: For each of the largestn− |q|letters ofu (in decreasing order),

drop this letterdown andadd 1 to it;

moveit left until hitting some unoccupied sitei ∈/ q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7 4 4 5

Phase I: For each of the largestn− |q|letters ofu (in decreasing order),

drop this letterdown andadd 1 to it;

moveit left until hitting some unoccupied sitei ∈/ q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7 4 4 5 1

Phase II: For each of the smallest|q|letters ofu (in increasing order),

drop this letter down;

moveitright until hitting some unoccupied sitei ∈q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

7 7 4 4 5 1 1

Phase II: For each of the smallest|q|letters ofu (in increasing order),

drop this letter down;

moveitright until hitting some unoccupied sitei ∈q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

2 7 7 4 4 5 1 1

Phase II: For each of the smallest|q|letters ofu (in increasing order),

drop this letter down;

moveitright until hitting some unoccupied sitei ∈q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

2 7 7 3 4 4 5 1 1

Phase II: For each of the smallest|q|letters ofu (in increasing order),

drop this letter down;

moveitright until hitting some unoccupied sitei ∈q;

place it there. _{4 / 23}

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

2 7 7 3 4 4 5 1 1

The letters on the bottom now formq(u).

4 / 23

Action of queues on words, 1: example

Let q be a queue, andu a word. We shall define a wordq(u).

Idea:

Drawu on top.

Drawq as circles in the middle.

Buildq(u) letter by letter, as follows...

Example: n= 9 andu = 346613321 and q={1,4,8,9}:

3 4 6 6 1 3 3 2 1

2 7 7 3 4 4 5 1 1

The letters on the bottom now formq(u).

Proposition. Equal letters can be processed in any order.

4 / 23

Action of queues on words, 2: formal definition

Let q be a queue, andu a word. Define a word q(u) as follows:

In the beginning, v=q(u) is a word whose letters are unset.

Choose a permutation (i₁,i₂, . . . ,i_n) of (1,2, . . . ,n) such that ui1 ≤ui2 ≤ · · · ≤uin.

Phase I. For i =in,in−1, . . . ,i_|q|+1, do the following:

Find the first site j weakly to the left (cyclically) of i such that j ∈/q andvj is not set. Then set v_j =u_i+ 1.

Phase II. For i =i₁,i₂, . . . ,i_|q|, do the following:

Find the first site j weakly to the right

(cyclically) of i such that j ∈q andv_j is not set.

Then set v_j =u_i. Proposition.

The resulting word v =q(u) does not depend on the choice of permutation (i₁,i₂, . . . ,i_n).

Phase I and Phase II can be done in parallel.

5 / 23

Action of queues on words, 2: formal definition

Let q be a queue, andu a word. Define a word q(u) as follows:

In the beginning, v=q(u) is a word whose letters are unset.

Choose a permutation (i₁,i₂, . . . ,i_n) of (1,2, . . . ,n) such that ui1 ≤ui2 ≤ · · · ≤uin.

Phase I. For i =in,in−1, . . . ,i_|q|+1, do the following:

Find the first site j weakly to the left (cyclically) of i such that j ∈/q andvj is not set. Then set v_j =u_i+ 1.

Phase II. For i =i₁,i₂, . . . ,i_|q|, do the following:

Find the first site j weakly to the right

(cyclically) of i such that j ∈q andv_j is not set.

Then set v_j =u_i. Proposition.

The resulting word v =q(u) does not depend on the choice of permutation (i₁,i₂, . . . ,i_n).

Phase I and Phase II can be done in parallel.

5 / 23

Remark on TASEP connection

This action of queues on words is inspired by the “discrete MLQs” of Aas and Linusson (arXiv:1501.04417).

Main difference: They have no Phase I, but their words have empty positions.

(Our picture subsumes theirs – fill the empty positions with high letters.)

Their motivation: compute stationary distribution of TASEP (totally asymmetric exclusion process) on a circle.

Our work proves two of their conjectures.

6 / 23

Remark on TASEP connection

This action of queues on words is inspired by the “discrete MLQs” of Aas and Linusson (arXiv:1501.04417).

Main difference: They have no Phase I, but their words have empty positions.

(Our picture subsumes theirs – fill the empty positions with high letters.)

Their motivation: compute stationary distribution of TASEP (totally asymmetric exclusion process) on a circle.

Our work proves two of their conjectures.

6 / 23

Remark on TASEP connection

This action of queues on words is inspired by the “discrete MLQs” of Aas and Linusson (arXiv:1501.04417).

Main difference: They have no Phase I, but their words have empty positions.

(Our picture subsumes theirs – fill the empty positions with high letters.)

Their motivation: compute stationary distribution of TASEP (totally asymmetric exclusion process) on a circle.

Our work proves two of their conjectures.

6 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3,0,0,0, . . .).

We omit trailing zeroes from infinite sequences. That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0. Example: The word 1255135 is not packed. The word 1244134 is packed with 4 classes.

7 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3,0,0,0, . . .).

We omit trailing zeroes from infinite sequences.

That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0. Example: The word 1255135 is not packed. The word 1244134 is packed with 4 classes.

7 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3,0,0,0, . . .).

We omit trailing zeroes from infinite sequences.

That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0. Example: The word 1255135 is not packed. The word 1244134 is packed with 4 classes.

7 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3).

We omit trailing zeroes from infinite sequences.

That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0.

Example: The word 1255135 is not packed. The word 1244134 is packed with 4 classes.

7 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3).

We omit trailing zeroes from infinite sequences.

That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0.

Example: The word 1255135 is not packed.

The word 1244134 is packed with 4 classes.

7 / 23

Types of words

The typeof a word u is the sequence m= (m₁,m₂, . . .), wheremk = (# of all sites i such thatui =k).

Example: The word 1255135 has type (2,1,1,0,3).

We omit trailing zeroes from infinite sequences.

That is, we abbreviate (m1,m2, . . . ,mk,0,0,0, . . .) as (m1,m2, . . . ,m_k).

A word u is packed with`classes if its typem has m1,m2, . . . ,m` >0 and m`+1=m`+2=· · ·= 0.

Example: The word 1255135 is not packed.

The word 1244134 is packed with 4 classes.

7 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers. A σ-twisted MLQ of typem means an MLQ

q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m1+m2+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ

q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Example: n= 6 andm= (2,3,1) and`= 3 andσ = (2,1) (one-line notation). Then, a σ-twisted MLQ of typem is an MLQ q= (q1,q2) with|q₁|=m1+m2 = 2 + 3 = 5 and

|q₂|=m₁= 2.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Example: n= 6 andm= (2,3,1) and`= 3 andσ = (2,1) (one-line notation). Then, a σ-twisted MLQ of typem is an MLQ q= (q1,q2) with|q₁|=m1+m2 = 2 + 3 = 5 and

|q₂|=m₁= 2. For example,q= ({1,3,4,5,6},{2,3})

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Example: n= 6 andm= (2,3,1) and`= 3 andσ = (2,1) (one-line notation). Then, a σ-twisted MLQ of typem is an MLQ q= (q1,q2) with|q₁|=m1+m2 = 2 + 3 = 5 and

|q₂|=m₁= 2. For example,q= ({1,3,4,5,6},{2,3}) andq(111111) = 311222.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Example: n= 6 andm= (2,3,1) and`= 3 andσ = (2,1) (one-line notation). Then, a σ-twisted MLQ of typem is an MLQ q= (q1,q2) with|q₁|=m1+m2 = 2 + 3 = 5 and

|q₂|=m₁= 2. For example,q= ({1,3,4,5,6},{4,5})

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Example: n= 6 andm= (2,3,1) and`= 3 andσ = (2,1) (one-line notation). Then, a σ-twisted MLQ of typem is an MLQ q= (q1,q2) with|q₁|=m1+m2 = 2 + 3 = 5 and

|q₂|=m₁= 2. For example,q= ({1,3,4,5,6},{4,5}) andq(111111) = 232112.

8 / 23

MLQs

A MLQ(short for “multiline queue”) is a tuple of queues.

Ifq= (q1,q2, . . . ,qk) is an MLQ, andu is a word, then q(u) :=q_k(q_k−1(· · ·(q₁(u)))).

Let ` >0, and letσ be a permutation of [`−1].

Let m= (m1,m2, . . . ,m`) be a sequence of positive integers.

A σ-twisted MLQ of typem means an MLQ q= (q₁,q₂, . . . ,q`−1) such that

|q_i|=m₁+m₂+· · ·+m_σ(i) for all i, and n=m1+m2+· · ·.

Equivalently: Aσ-twisted MLQ of type mcan be defined as an MLQ q= (q1,q2, . . . ,q`−1) such that

the wordq(1· · ·1) has type m (where 1· · ·1 is the word whose values all equal 1);

we have 0<

q_σ⁻¹₍₁₎ <

q_σ⁻¹₍₂₎

<· · ·<

q_σ⁻¹_(`−1) .

8 / 23

Generating functions, 1: definition

Now, letx₁,x₂, . . . ,x_n be commuting variables.

For any`≥1, any permutationσ of [`−1], and any packed word u of typem with `classes, we define theσ-spectral weight hui_σ by

hui_σ := X

qis aσ-twisted MLQ of typem satisfyingu=q(1···1)

wtq.

Here:

1· · ·1 denotes the word whose all values are 1.

wtq:=

k

Q

p=1

Q

i∈qp

xi for any MLQ q= (q1,q2, . . . ,qk).

9 / 23

Generating functions, 1: definition

Now, letx₁,x₂, . . . ,x_n be commuting variables.

For any`≥1, any permutationσ of [`−1], and any packed word u of typem with `classes, we define theσ-spectral weight hui_σ by

hui_σ := X

qis aσ-twisted MLQ of typem satisfyingu=q(1···1)

wtq.

Here:

1· · ·1 denotes the word whose all values are 1.

wtq:=

k

Q

p=1

Q

i∈qp

xi for any MLQ q= (q1,q2, . . . ,qk).

9 / 23

Generating functions, 1: definition

Now, letx₁,x₂, . . . ,x_n be commuting variables.

For any`≥1, any permutationσ of [`−1], and any packed word u of typem with `classes, we define theσ-spectral weight hui_σ by

hui_σ := X

qis aσ-twisted MLQ of typem satisfyingu=q(1···1)

wtq.

Here:

1· · ·1 denotes the word whose all values are 1.

wtq:=

k

Q

p=1

Q

i∈qp

xi for any MLQ q= (q1,q2, . . . ,qk).

Example: Recall that ({1,3,4,5,6},{4,5}) is a σ-twisted MLQ of typem for n= 6 andm= (2,3,1) and `= 3 and σ = (2,1) (one-line notation) satisfyingq(111111) = 232112.

It contributes a monomial

(x₁x₃x₄x₅x₆) (x₄x₅) =x₁x₃x₄²x₅²x₆ to h232112i_σ.

9 / 23

Generating functions, 1: definition

Now, letx₁,x₂, . . . ,x_n be commuting variables.

For any`≥1, any permutationσ of [`−1], and any packed word u of typem with `classes, we define theσ-spectral weight hui_σ by

hui_σ := X

qis aσ-twisted MLQ of typem satisfyingu=q(1···1)

wtq.

Here:

1· · ·1 denotes the word whose all values are 1.

wtq:=

k

Q

p=1

Q

i∈qp

xi for any MLQ q= (q1,q2, . . . ,qk).

Set hui:=hui_id for the permutation id of [`−1].

9 / 23

Generating functions, 2: more examples

Example: For n= 5,`= 5 andm= (1,1,2,1), we have h13234i=x₁x₂x₃²x₄(x₁²+x₁x₄+x₁x₅+x₄x₅+x₅²).

Examples: For n= 5, `= 5 andm= (1,1,1,1,1), we have h13245i=x1x2x₃²x4(x₁²+x1x4+x1x5+x₄²+x4x5+x₅²)

·(x1x2x3+x1x2x5+x1x3x5+x2x3x5),

h14235i=x₁x₂x₃²x₄²(x₁³x₂+x₁³x₃+x₁³x₅+x₁²x₂x₃+x₁²x₂x₄ + 2x₁²x2x5+x₁²x3x4+ 2x₁²x3x5+x₁²x4x5

+x₁²x₅²+x1x2x3x5+x1x2x4x5+ 2x1x2x₅² +x₁x₃x₄x₅+ 2x₁x₃x₅²+x₁x₄x₅²+x₁x₅³ +x2x3x₅²+x2x4x₅²+x2x₅³+x3x4x₅²+x3x₅³).

10 / 23

Generating functions, 2: more examples

Example: For n= 5,`= 5 andm= (1,1,2,1), we have h13234i=x₁x₂x₃²x₄(x₁²+x₁x₄+x₁x₅+x₄x₅+x₅²).

Examples: For n= 5, `= 5 andm= (1,1,1,1,1), we have h13245i=x1x2x₃²x4(x₁²+x1x4+x1x5+x₄²+x4x5+x₅²)

·(x1x2x3+x1x2x5+x1x3x5+x2x3x5),

h14235i=x₁x₂x₃²x₄²(x₁³x₂+x₁³x₃+x₁³x₅+x₁²x₂x₃+x₁²x₂x₄ + 2x₁²x2x5+x₁²x3x4+ 2x₁²x3x5+x₁²x4x5

+x₁²x₅²+x1x2x3x5+x1x2x4x5+ 2x1x2x₅² +x₁x₃x₄x₅+ 2x₁x₃x₅²+x₁x₄x₅²+x₁x₅³ +x2x3x₅²+x2x4x₅²+x2x₅³+x3x4x₅²+x3x₅³).

10 / 23

The symmetry theorem, 1: statement

Theorem. For any`≥1, any permutationσ of [`−1], and any packed word u of type mwith ` classes, we have

hui_σ =hui.

This yields a recent conjecture by Arita, Ayyer, Mallick and Prolhac on the TASEP.

This is proven bijectively, using a “duality transformation” on MLQs that leaves their action on words unchanged.

Main lemma. Ifq1 andq2 are two queues, then there are two queues q₁⁰ andq₂⁰ satisfying

q₁⁰

=|q₂| and q⁰₂

=|q₁| and



 Y

i∈q⁰₁

x_i







 Y

i∈q⁰₂

x_i



=



 Y

i∈q1

x_i







 Y

i∈q2

x_i





such that every wordu satisfies q₁⁰ q₂⁰ (u)

=q1(q2(u)).

11 / 23

The symmetry theorem, 1: statement

Theorem. For any`≥1, any permutationσ of [`−1], and any packed word u of type mwith ` classes, we have

hui_σ =hui.

This yields a recent conjecture by Arita, Ayyer, Mallick and Prolhac on the TASEP.

This is proven bijectively, using a “duality transformation” on MLQs that leaves their action on words unchanged.

Main lemma. Ifq1 andq2 are two queues, then there are two queues q₁⁰ andq₂⁰ satisfying

q₁⁰

=|q₂| and q₂⁰

=|q₁| and



 Y

i∈q⁰₁

x_i







 Y

i∈q⁰₂

x_i



=



 Y

i∈q1

x_i







 Y

i∈q2

x_i





such that every wordu satisfies q₁⁰ q₂⁰ (u)

=q1(q2(u)).

11 / 23

The symmetry theorem, 1: statement

Theorem. For any`≥1, any permutationσ of [`−1], and any packed word u of type mwith ` classes, we have

hui_σ =hui.

This yields a recent conjecture by Arita, Ayyer, Mallick and Prolhac on the TASEP.

This is proven bijectively, using a “duality transformation” on MLQs that leaves their action on words unchanged.

Main lemma. Ifq1 andq2 are two queues, then there are two queues q₁⁰ andq₂⁰ satisfying

q₁⁰

=|q₂| and q₂⁰

=|q₁| and



 Y

i∈q⁰₁

x_i







 Y

i∈q⁰₂

x_i



=



 Y

i∈q1

x_i







 Y

i∈q2

x_i





such that every wordu satisfies q₁⁰ q₂⁰ (u)

=q1(q2(u)).

11 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Convenient example:

n= 10;

q1={2,6,7,9}; q2={1,3,5,7,8}.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Convenient example:

n= 10;

q1={2,6,7,9}; q2={1,3,5,7,8}. Then,

b = ◦) (◦ ◦) ◦◦ ◦) (◦ () ◦) (◦ ◦◦

i = 1 2 3 4 5 6 7 8 9 10

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Convenient example:

n= 10;

q1={2,6,7,9}; q2={1,3,5,7,8}. Then,

b = ◦) (◦ ◦) ◦◦ ◦) (◦ () ◦) (◦ ◦◦

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Convenient example:

n= 10;

q1={2,6,7,9}; q2={1,3,5,7,8}. Then,

b=◦)(◦◦)◦◦◦)(◦()◦)(◦◦◦

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically. In our above example:

b=◦)(◦◦)◦◦◦)(◦()◦)(◦◦◦

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically. In our above example:

b=◦)(₁◦◦)₁◦◦◦)(◦()◦)(◦◦◦

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically. In our above example:

b=◦)(₁◦◦)₁◦◦◦)(◦(₂)₂◦)(◦◦◦

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically. In our above example:

b=◦)(₁◦◦)₁◦◦◦)(₃◦(₂)2◦)₃(◦◦◦

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically. In our above example:

b =◦)₄(1◦◦)₁◦◦◦)(₃◦(₂)2◦)₃(4◦◦◦

Replace the unmatched parentheses by their duals – e.g., if they were )’s, make them (’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23

The symmetry theorem, 2: idea of proof

The construction of q₁⁰ andq₂⁰ is combinatorial:

Encode the pair (q1,q2) as a 2n-letter word

b= (b1,b2, . . . ,b2n) over the 3-letter alphabet{),(,◦}.

Namely, for each i,

let b2i−1 be an opening parenthesis “(” if i ∈q₁, otherwise a neutral symbol “◦”;

let b_2i be a closing parenthesis “)” if i ∈q₂, otherwise a neutral symbol “◦”.

Match parentheses inb “the usual way” but keeping in mind that the word wraps around cyclically.

Replace the unmatched parentheses by their duals – e.g., if they were)’s, make them(’s.

In our above example:

b =◦)₄(₁◦◦)₁◦◦◦)(₃◦(₂)₂◦)₃(₄◦◦◦

Turn the resulting wordb⁰ into two sets q₁⁰ andq₂⁰ as follows:

q₁⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “(” ; q₂⁰ =

i ∈[n] | either b⁰_2i−1 or b_2i⁰ is a “)” .

12 / 23