Institut Mittag-Leffler, Djursholm 2020

The Petrie symmetrie functions and Murnaghan–Nakayama rules

Darij Grinberg

4 February 2020

Institut Mittag–Leffler, Djursholm, Sweden

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

djursholm2020.pdf paper: http:

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

//www.cip.ifi.lmu.de/~grinberg/algebra/fps20pet.pdf

1 / 43

Manifest

What you are going to see:

A new family (G(k,m))_m≥0 of symmetric functions for eachk >0. (So, a family of families.)

It “interpolates” between thee’s and the h’s in a sense.

Various nice properties if I do say so myself.

A proof (sketch) of a conjecture coming from algebraic groups.

A source of homework exercises for your symmetric functions class.

What you arenot going to see:

Meaning.

Theories.

(mostly) actual combinatorics (algorithms, bijections, etc.).

2 / 43

Manifest

What you are going to see:

A new family (G(k,m))_m≥0 of symmetric functions for eachk >0. (So, a family of families.)

It “interpolates” between thee’s and the h’s in a sense.

Various nice properties if I do say so myself.

A proof (sketch) of a conjecture coming from algebraic groups.

A source of homework exercises for your symmetric functions class.

What you arenot going to see:

Meaning.

Theories.

(mostly) actual combinatorics (algorithms, bijections, etc.).

2 / 43

Symmetric functions: notation, 1

We will use standard notations for symmetric functions, such as used in:

Richard Stanley, Enumerative Combinatorics, volume 2, CUP 2001.

D.G. and Victor Reiner,Hopf algebras in Combinatorics, 2012-2020+.

Let k be a commutative ring (ZandQ will suffice).

Let N:={0,1,2, . . .}.

3 / 43

Symmetric functions: notation, 1

We will use standard notations for symmetric functions, such as used in:

Richard Stanley, Enumerative Combinatorics, volume 2, CUP 2001.

D.G. and Victor Reiner,Hopf algebras in Combinatorics, 2012-2020+.

Let k be a commutative ring (ZandQ will suffice).

Let N:={0,1,2, . . .}.

3 / 43

Symmetric functions: notation, 2

A weak composition means a sequence (α1, α2, α3, . . .)∈N^∞ such that all i 0 satisfy α_i = 0.

We let WCbe the set of all weak compositions.

We writeα_i for the i-th entry of a weak compositionα.

The sizeof a weak compositionα is defined to be

|α|:=α₁+α₂+α₃+· · ·.

A partitionmeans a weak compositionα satisfying α₁≥α₂ ≥α₃ ≥ · · ·.

A partition of n means a partitionα with |α|=n.

We let Par denote the set of all partitions. For eachn ∈Z, we let Par_n denote the set of all partitions of n.

We often omit trailing zeroes from partitions: e.g., (3,2,1,0,0,0, . . .) = (3,2,1) = (3,2,1,0).

The partition (0,0,0, . . .) = () is called theempty partition and denoted by ∅.

4 / 43

Symmetric functions: notation, 2

A weak composition means a sequence (α1, α2, α3, . . .)∈N^∞ such that all i 0 satisfy α_i = 0.

We let WCbe the set of all weak compositions.

We writeα_i for the i-th entry of a weak compositionα.

The sizeof a weak compositionα is defined to be

|α|:=α₁+α₂+α₃+· · ·.

A partitionmeans a weak compositionα satisfying α₁≥α₂ ≥α₃ ≥ · · ·.

A partition of n means a partitionα with |α|=n.

We letPardenote the set of all partitions. For eachn ∈Z, we let Par_n denote the set of all partitions of n.

We often omit trailing zeroes from partitions: e.g., (3,2,1,0,0,0, . . .) = (3,2,1) = (3,2,1,0).

The partition (0,0,0, . . .) = () is called theempty partition and denoted by ∅.

4 / 43

Symmetric functions: notation, 2

A weak composition means a sequence (α1, α2, α3, . . .)∈N^∞ such that all i 0 satisfy α_i = 0.

We let WCbe the set of all weak compositions.

We writeα_i for the i-th entry of a weak compositionα.

The sizeof a weak compositionα is defined to be

|α|:=α₁+α₂+α₃+· · ·.

A partitionmeans a weak compositionα satisfying α₁≥α₂ ≥α₃ ≥ · · ·.

A partition of n means a partitionα with |α|=n.

We letPardenote the set of all partitions. For eachn ∈Z, we let Par_n denote the set of all partitions of n.

We often omit trailing zeroes from partitions: e.g., (3,2,1,0,0,0, . . .) = (3,2,1) = (3,2,1,0).

The partition (0,0,0, . . .) = () is called theempty partition and denoted by ∅.

4 / 43

Symmetric functions: notation, 3

We will use the notationm^k for “m,m, . . . ,m

| {z }

ktimes

” in partitions.

(For example, 2,1⁴

= (2,1,1,1,1).)

For any weak composition α, we let x^α denote the monomial x₁^α1x₂^α2x₃^α3· · ·. It has degree|α|.

The ring k [[x₁,x₂,x₃, . . .]] consists of formal infinite k-linear combinations of monomials x^α. These combinations are called formal power series.

The symmetric functionsare the formal power series f ∈k [[x1,x2,x3, . . .]] that are

of bounded degree (i.e., all monomials in f have degrees

<N for someN=N_f);

symmetric(i.e., permuting the xi does not changef). We let

Λ={symmetric functions f ∈k [[x1,x2,x3, . . .]]}. This is a k-subalgebra of k [[x1,x2,x3, . . .]], graded by the degree.

5 / 43

Symmetric functions: notation, 3

We will use the notationm^k for “m,m, . . . ,m

| {z }

ktimes

” in partitions.

(For example, 2,1⁴

= (2,1,1,1,1).)

For any weak composition α, we letx^α denote the monomial x₁^α1x₂^α2x₃^α3· · ·. It has degree|α|.

The ring k [[x1,x2,x3, . . .]] consists of formal infinite k-linear combinations of monomials x^α. These combinations are called formal power series.

The symmetric functionsare the formal power series f ∈k [[x1,x2,x3, . . .]] that are

of bounded degree (i.e., all monomials in f have degrees

<N for someN=N_f);

symmetric(i.e., permuting the xi does not changef).

We let

Λ={symmetric functions f ∈k [[x1,x2,x3, . . .]]}. This is a k-subalgebra of k [[x1,x2,x3, . . .]], graded by the degree.

5 / 43

Symmetric functions: notation, 3

We will use the notationm^k for “m,m, . . . ,m

| {z }

ktimes

” in partitions.

(For example, 2,1⁴

= (2,1,1,1,1).)

For any weak composition α, we letx^α denote the monomial x₁^α1x₂^α2x₃^α3· · ·. It has degree|α|.

The ring k [[x1,x2,x3, . . .]] consists of formal infinite k-linear combinations of monomials x^α. These combinations are called formal power series.

The symmetric functionsare the formal power series f ∈k [[x1,x2,x3, . . .]] that are

of bounded degree (i.e., all monomials in f have degrees

<N for someN=N_f);

symmetric(i.e., permuting the xi does not changef).

We let

Λ ={symmetric functions f ∈k [[x1,x2,x3, . . .]]}. This is a k-subalgebra of k [[x1,x2,x3, . . .]], graded by the degree.

5 / 43

Symmetric functions: notation, 3

We will use the notationm^k for “m,m, . . . ,m

| {z }

ktimes

” in partitions.

(For example, 2,1⁴

= (2,1,1,1,1).)

For any weak composition α, we letx^α denote the monomial x₁^α1x₂^α2x₃^α3· · ·. It has degree|α|.

The ring k [[x1,x2,x3, . . .]] consists of formal infinite k-linear combinations of monomials x^α. These combinations are called formal power series.

The symmetric functionsare the formal power series f ∈k [[x1,x2,x3, . . .]] that are

of bounded degree (i.e., all monomials in f have degrees

<N for someN=N_f);

symmetric(i.e., permuting the xi does not changef).

We let

Λ={symmetric functions f ∈k [[x1,x2,x3, . . .]]}. This is a k-subalgebra of k [[x1,x2,x3, . . .]], graded by the degree.

5 / 43

Symmetric functions: m

The k-module Λ has several bases indexed by the set Par.

The monomial basis(m_λ)_λ∈Par:

For each partitionλ, we define themonomial symmetric function mλ ∈Λ by

m_λ = X

αis a weak composition;

αcan be obtained fromλ by permuting entries

x^α.

For example: m_(2,2,1)= X

i<j<k

x_i²x_j²x_k+ X

i<j<k

x_i²x_jx_k²+ X

i<j<k

x_ix_j²x_k².

The family (m_λ)_λ∈Par is a basis of the k-module Λ, called the monomial basis.

6 / 43

Symmetric functions: m

The k-module Λ has several bases indexed by the set Par.

The monomial basis(m_λ)_λ∈Par:

For each partitionλ, we define themonomial symmetric functionmλ ∈Λ by

m_λ = X

αis a weak composition;

αcan be obtained fromλ by permuting entries

x^α.

For example:

m_(2,2,1)= X

i<j<k

x_i²x_j²x_k+ X

i<j<k

x_i²x_jx_k²+ X

i<j<k

x_ix_j²x_k².

The family (m_λ)_λ∈Par is a basis of the k-module Λ, called the monomial basis.

6 / 43

Symmetric functions: m

The k-module Λ has several bases indexed by the set Par.

The monomial basis(m_λ)_λ∈Par:

For each partitionλ, we define themonomial symmetric functionmλ ∈Λ by

m_λ = X

αis a weak composition;

αcan be obtained fromλ by permuting entries

x^α.

For example:

m_(2,2,1)= X

i<j<k

x_i²x_j²x_k+ X

i<j<k

x_i²x_jx_k²+ X

i<j<k

x_ix_j²x_k².

The family (m_λ)_λ∈Par is a basis of the k-module Λ, called the monomial basis.

6 / 43

Symmetric functions: m

The k-module Λ has several bases indexed by the set Par.

The monomial basis(m_λ)_λ∈Par:

For each partitionλ, we define themonomial symmetric functionmλ ∈Λ by

m_λ = X

αis a weak composition;

αcan be obtained fromλ by permuting entries

x^α.

For example:

m_(2,2,1)= X

i<j<k

x_i²x_j²x_k+ X

i<j<k

x_i²x_jx_k²+ X

i<j<k

x_ix_j²x_k².

The family (m_λ)_λ∈Par is a basis of the k-module Λ, called the monomial basis.

6 / 43

Symmetric functions: h

The complete basis (h_λ)_λ∈Par:

For eachn∈Z, define the complete homogeneous symmetric function h_n by

h_n = X

i1≤i2≤···≤in

x_i1x_i2· · ·x_in = X

α∈WC;

|α|=n

x^α = X

λ∈Parn

m_λ.

For example,

h₁ =x₁+x₂+x₃+· · · ; h₂ =X

i≤j

x_ix_j =X

i

x_i²+X

i<j

x_ix_j; h0 = 1;

h_n= 0 for all n<0.

For each partitionλ, we define

h_λ =h_λ1h_λ2h_λ3· · · ∈Λ. The family (hλ)_λ∈Par is a basis of the k-module Λ.

7 / 43

Symmetric functions: h

The complete basis (h_λ)_λ∈Par:

For eachn∈Z, define the complete homogeneous symmetric function h_n by

h_n = X

i1≤i2≤···≤in

x_i1x_i2· · ·x_in = X

α∈WC;

|α|=n

x^α = X

λ∈Parn

m_λ.

For example,

h₁ =x₁+x₂+x₃+· · · ; h₂ =X

i≤j

x_ix_j =X

i

x_i²+X

i<j

x_ix_j; h0 = 1;

h_n= 0 for all n<0.

For each partitionλ, we define

h_λ =h_λ1h_λ2h_λ3· · · ∈Λ.

The family (hλ)_λ∈Par is a basis of the k-module Λ.

7 / 43

Symmetric functions: h

The complete basis (h_λ)_λ∈Par:

For eachn∈Z, define the complete homogeneous symmetric function h_n by

h_n = X

i1≤i2≤···≤in

x_i1x_i2· · ·x_in = X

α∈WC;

|α|=n

x^α = X

λ∈Parn

m_λ.

For example,

h₁ =x₁+x₂+x₃+· · · ; h₂ =X

i≤j

x_ix_j =X

i

x_i²+X

i<j

x_ix_j; h0 = 1;

h_n= 0 for all n<0.

For each partitionλ, we define

h_λ =h_λ1h_λ2h_λ3· · · ∈Λ.

The family (hλ)_λ∈Par is a basis of the k-module Λ.

7 / 43

Symmetric functions: e

The elementary basis(e_λ)_λ∈Par:

For eachn∈Z, define the elementary symmetric function en

by

e_n= X

i1<i2<···<in

x_i1x_i2· · ·x_in = X

α∈WC∩{0,1}^∞;

|α|=n

x^α=m₍₁ⁿ₎.

For example,

e1 =x1+x2+x3+· · · ; e2 =X

i<j

xixj; e0 = 1;

en= 0 for all n<0.

For each partitionλ, we define

e_λ=e_λ1e_λ2e_λ3· · · ∈Λ.

The family (e_λ)_λ∈Par is a basis of the k-module Λ.

8 / 43

Symmetric functions: e

The elementary basis(e_λ)_λ∈Par:

For eachn∈Z, define the elementary symmetric function en

by

e_n= X

i1<i2<···<in

x_i1x_i2· · ·x_in = X

α∈WC∩{0,1}^∞;

|α|=n

x^α=m₍₁ⁿ₎.

For example,

e1 =x1+x2+x3+· · · ; e2 =X

i<j

xixj; e0 = 1;

en= 0 for all n<0.

For each partitionλ, we define

e_λ=e_λ1e_λ2e_λ3· · · ∈Λ.

The family (e_λ)_λ∈Par is a basis of the k-module Λ.

8 / 43

Symmetric functions: e

The elementary basis(e_λ)_λ∈Par:

For eachn∈Z, define the elementary symmetric function en

by

e_n= X

i1<i2<···<in

x_i1x_i2· · ·x_in = X

α∈WC∩{0,1}^∞;

|α|=n

x^α=m₍₁ⁿ₎.

For example,

e1 =x1+x2+x3+· · · ; e2 =X

i<j

xixj; e0 = 1;

en= 0 for all n<0.

For each partitionλ, we define

e_λ=e_λ1e_λ2e_λ3· · · ∈Λ.

The family (e_λ)_λ∈Par is a basis of the k-module Λ.

8 / 43

Symmetric functions: p

The power-sum symmetric functionsp_n:

For each positive integer n, define the power-sum symmetric function p_n by

pn=x₁ⁿ+x₂ⁿ+x₃ⁿ+· · ·=m_(n).

We can make a basis out of (products of) pn’s when k is a Q-algebra.

9 / 43

Symmetric functions: p

The power-sum symmetric functionsp_n:

For each positive integer n, define the power-sum symmetric function p_n by

pn=x₁ⁿ+x₂ⁿ+x₃ⁿ+· · ·=m_(n).

We can make a basis out of (products of) pn’s when k is a Q-algebra.

9 / 43

Symmetric functions: s

The Schur basis(sλ)_λ∈Par:

For each partitionλ, we can define theSchur function s_λ in many equivalent ways, e.g.:

We have

s_λ = X

T is a semistandard Young tableau of shapeλ

x_T,

where xT denotes the monomial obtained by multiplying thexi for all entriesi of T.

Ifλ= (λ₁, λ₂, . . . , λ_`), then s_λ= det

(h_λi−i+j)_{1≤i≤`,1≤j≤`}

(thefirst Jacobi–Trudi formula).

The family (s_λ)_λ∈Par is a basis of the k-module Λ.

10 / 43

Symmetric functions: s

The Schur basis(sλ)_λ∈Par:

For each partitionλ, we can define theSchur function s_λ in many equivalent ways, e.g.:

We have

s_λ = X

T is a semistandard Young tableau of shapeλ

x_T,

where xT denotes the monomial obtained by multiplying thexi for all entriesi of T.

Ifλ= (λ₁, λ₂, . . . , λ_`), then s_λ= det

(h_λi−i+j)_{1≤i≤`,1≤j≤`}

(thefirst Jacobi–Trudi formula).

The family (s_λ)_λ∈Par is a basis of the k-module Λ.

10 / 43

Petrie functions: definition of G(k) For any positive integer k, set

G(k)

= X

α∈WC;

αi<kfor alli

x^α

=X

(all monomials whose exponents are all <k)

∈k [[x1,x2,x3, . . .]] (not ∈Λ in general).

11 / 43

Petrie functions: definition of G(k,m)

For any positive integer k and any m∈N, we let G(k,m)

= X

α∈WC;

|α|=m;

αi<kfor alli

x^α

=X

(all degree-m monomials whose exponents are all <k)

∈Λ.

For example, G(3,4) = X

i<j<k<`

xixjxkx`+ X

i<j<k

x_i²xjxk + X

i<j<k

xix_j²xk

+ X

i<j<k

xixjx_k²+X

i<j

x_i²x_j²

=m_(1,1,1,1)+m_(2,1,1)+m_(2,2).

12 / 43

Petrie functions: definition of G(k,m)

For any positive integer k and any m∈N, we let G(k,m)

= X

α∈WC;

|α|=m;

αi<kfor alli

x^α

=X

(all degree-m monomials whose exponents are all <k)

∈Λ.

For example, G(3,4) = X

i<j<k<`

xixjxkx`+ X

i<j<k

x_i²xjxk + X

i<j<k

xix_j²xk

+ X

i<j<k

xixjx_k²+X

i<j

x_i²x_j²

=m_(1,1,1,1)+m_(2,1,1)+m_(2,2).

12 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k). G(k,m) = X

λ∈Par;

|λ|=m; λi<kfor alli

mλ.

G(2,m) =e_m.

G(k,m) =h_m whenever k>m. G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m; λi<kfor alli

mλ.

G(2,m) =e_m.

G(k,m) =h_m whenever k>m. G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m;

λi<kfor alli

m_λ.

G(2,m) =e_m.

G(k,m) =h_m whenever k>m. G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m;

λi<kfor alli

m_λ.

G(2,m) =e_m.

G(k,m) =h_m whenever k>m. G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m;

λi<kfor alli

m_λ.

G(2,m) =e_m.

G(k,m) =h_m wheneverk >m.

G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m;

λi<kfor alli

m_λ.

G(2,m) =e_m.

G(k,m) =h_m wheneverk >m.

G(m,m) =hm−pm.

13 / 43

Petrie functions: basic properties

I named G(k) andG(k,m) the Petrie functions, for reasons that will become clear eventually.

Basic properties(for arbitraryk >0 andm∈N):

G(k) = X

λ∈Par;

λi<kfor alli

mλ =

∞

Y

i=1

x_i⁰+x_i¹+· · ·+x_i^k−1

.

G(k,m) is them-th degree component ofG(k).

G(k,m) = X

λ∈Par;

|λ|=m;

λi<kfor alli

m_λ.

G(2,m) =e_m.

G(k,m) =h_m wheneverk >m.

G(m,m) =hm−pm.

13 / 43

Petrie functions and the coproduct of Λ This is for the friends of Hopf algebras:

∆ (G(k,m)) =

m

X

i=0

G(k,i)⊗G(k,m−i) for eachk >0 andm∈N.

Here, ∆is the comultiplicationof Λ, defined to be the k-algebra homomorphism

∆ : Λ→Λ⊗Λ, en7→

n

X

i=0

ei ⊗en−i.

In terms of alphabets, this says

(G(k,m)) (x₁,x₂,x₃, . . . ,y₁,y₂,y₃, . . .)

=

m

X

i=0

(G(k,i)) (x1,x2,x3, . . .)·(G(k,m−i)) (y1,y2,y3, . . .).

14 / 43

Petrie functions and the coproduct of Λ This is for the friends of Hopf algebras:

∆ (G(k,m)) =

m

X

i=0

G(k,i)⊗G(k,m−i) for eachk >0 andm∈N.

Here, ∆is the comultiplicationof Λ, defined to be the k-algebra homomorphism

∆ : Λ→Λ⊗Λ, en7→

n

X

i=0

ei ⊗en−i.

In terms of alphabets, this says

(G(k,m)) (x₁,x₂,x₃, . . . ,y₁,y₂,y₃, . . .)

=

m

X

i=0

(G(k,i)) (x1,x2,x3, . . .)·(G(k,m−i)) (y1,y2,y3, . . .).

14 / 43

Expanding Petries in the Schur basis

We can expand the G(k,m) in the Schur basis (s_λ)_λ∈Par: e.g., G(4,6) =s(2,1,1,1,1)−s_(2,2,1,1)+s_(3,3).

Surprisingly, it turns out that all coefficients are in {0,1,−1}.

Better yet: Any product G(k,m)·s_µ expands in the Schur basis with coefficients in{0,1,−1}.

Let us see what the coefficients are.

15 / 43

Expanding Petries in the Schur basis

We can expand the G(k,m) in the Schur basis (s_λ)_λ∈Par: e.g., G(4,6) =s(2,1,1,1,1)−s_(2,2,1,1)+s_(3,3).

Surprisingly, it turns out that all coefficients are in {0,1,−1}.

Better yet: Any product G(k,m)·s_µ expands in the Schur basis with coefficients in{0,1,−1}.

Let us see what the coefficients are.

15 / 43

Expanding Petries in the Schur basis

We can expand the G(k,m) in the Schur basis (s_λ)_λ∈Par: e.g., G(4,6) =s(2,1,1,1,1)−s_(2,2,1,1)+s_(3,3).

Surprisingly, it turns out that all coefficients are in {0,1,−1}.

Better yet: Any product G(k,m)·s_µ expands in the Schur basis with coefficients in{0,1,−1}.

Let us see what the coefficients are.

15 / 43

Expanding Petries in the Schur basis

We can expand the G(k,m) in the Schur basis (s_λ)_λ∈Par: e.g., G(4,6) =s(2,1,1,1,1)−s_(2,2,1,1)+s_(3,3).

Surprisingly, it turns out that all coefficients are in {0,1,−1}.

Better yet: Any product G(k,m)·s_µ expands in the Schur basis with coefficients in{0,1,−1}.

Let us see what the coefficients are.

15 / 43

Petrie numbers

We let [A] denote the truth valueof a statementA (that is, 1 if Ais true, and 0 ifA is false).

Let λ= (λ₁, λ₂, . . . , λ_`)∈Par and

µ= (µ₁, µ₂, . . . , µ_`)∈Par, and letk be a positive integer.

Then, the k-Petrie numberpet_k(λ, µ) ofλand µis the integer defined by

pet_k(λ, µ) = det

([0≤λ_i −µ_j −i+j <k])_{1≤i≤`,1≤j≤`}

.

Proposition: We have pet_k(λ, µ)∈ {0,1,−1} for all λand µ.

Proof idea. Each row of the matrix

([0≤λi −µj −i+j <k])_{1≤i≤`,1≤j≤`} has the form (0,0, . . . ,0

| {z }

azeroes

,1,1, . . . ,1

| {z }

bones

,0,0, . . . ,0

| {z }

czeroes

) for somea,b,c ∈N. Thus, this matrix is the transpose of a Petrie matrix. Hence, its determinant is ∈ {−1,0,1} (byGordon and Wilkinson 1974).

16 / 43

Petrie numbers

We let [A] denote the truth valueof a statementA (that is, 1 if Ais true, and 0 ifA is false).

Let λ= (λ₁, λ₂, . . . , λ_`)∈Par and

µ= (µ₁, µ₂, . . . , µ_`)∈Par, and letk be a positive integer.

Then, the k-Petrie numberpet_k(λ, µ) ofλand µis the integer defined by

pet_k(λ, µ) = det

([0≤λ_i −µ_j −i+j <k])_{1≤i≤`,1≤j≤`}

.

Proposition: We have pet_k(λ, µ)∈ {0,1,−1} for all λand µ.

Proof idea. Each row of the matrix

([0≤λi −µj −i+j <k])_{1≤i≤`,1≤j≤`} has the form (0,0, . . . ,0

| {z }

azeroes

,1,1, . . . ,1

| {z }

bones

,0,0, . . . ,0

| {z }

czeroes

) for somea,b,c ∈N. Thus, this matrix is the transpose of a Petrie matrix. Hence, its determinant is ∈ {−1,0,1} (byGordon and Wilkinson 1974).

16 / 43

Petrie numbers

We let [A] denote the truth valueof a statementA (that is, 1 if Ais true, and 0 ifA is false).

Let λ= (λ₁, λ₂, . . . , λ_`)∈Par and

µ= (µ₁, µ₂, . . . , µ_`)∈Par, and letk be a positive integer.

Then, the k-Petrie numberpet_k(λ, µ) ofλand µis the integer defined by

pet_k(λ, µ) = det

([0≤λ_i −µ_j −i+j <k])_{1≤i≤`,1≤j≤`}

. For example, for `= 3, we have

pet_k(λ, µ)

= det





[0≤λ1−µ1<k] [0≤λ1−µ2+ 1<k] [0≤λ1−µ3+ 2<k]

[0≤λ2−µ1−1<k] [0≤λ2−µ2<k] [0≤λ2−µ3+ 1<k]

[0≤λ3−µ1−2<k] [0≤λ3−µ2−1<k] [0≤λ3−µ3<k]



.

For example,

pet₄((3,1,1),(2,1)) = det





1 1 0 0 1 1 0 0 1



= 1.

Proposition: We have pet_k(λ, µ)∈ {0,1,−1} for all λand µ.

Proof idea. Each row of the matrix

([0≤λi −µj −i+j <k])_{1≤i≤`,1≤j≤`} has the form (0,0, . . . ,0

| {z }

azeroes

,1,1, . . . ,1

| {z }

bones

,0,0, . . . ,0

| {z }

czeroes

) for somea,b,c ∈N. Thus, this matrix is the transpose of a Petrie matrix. Hence, its determinant is ∈ {−1,0,1} (byGordon and Wilkinson 1974).

16 / 43

Petrie numbers

We let [A] denote the truth valueof a statementA (that is, 1 if Ais true, and 0 ifA is false).

Let λ= (λ₁, λ₂, . . . , λ_`)∈Par and

µ= (µ₁, µ₂, . . . , µ_`)∈Par, and letk be a positive integer.

Then, the k-Petrie numberpet_k(λ, µ) ofλand µis the integer defined by

pet_k(λ, µ) = det

([0≤λ_i −µ_j −i+j <k])_{1≤i≤`,1≤j≤`}

. Proposition: We have pet_k(λ, µ)∈ {0,1,−1} for all λand µ.

Proof idea. Each row of the matrix

([0≤λi−µj −i+j <k])_{1≤i≤`,1≤j≤`} has the form (0,0, . . . ,0

| {z }

azeroes

,1,1, . . . ,1

| {z }

bones

,0,0, . . . ,0

| {z }

czeroes

) for somea,b,c ∈N.

Thus, this matrix is the transpose of a Petrie matrix. Hence, its determinant is ∈ {−1,0,1} (by Gordon and Wilkinson 1974).

16 / 43

Petrie numbers

We let [A] denote the truth valueof a statementA (that is, 1 if Ais true, and 0 ifA is false).

Let λ= (λ₁, λ₂, . . . , λ_`)∈Par and

µ= (µ₁, µ₂, . . . , µ_`)∈Par, and letk be a positive integer.

Then, the k-Petrie numberpet_k(λ, µ) ofλand µis the integer defined by

pet_k(λ, µ) = det

([0≤λ_i −µ_j −i+j <k])_{1≤i≤`,1≤j≤`}

. Proposition: We have pet_k(λ, µ)∈ {0,1,−1} for all λand µ.

Proof idea. Each row of the matrix

([0≤λi−µj −i+j <k])_{1≤i≤`,1≤j≤`} has the form (0,0, . . . ,0

| {z }

azeroes

,1,1, . . . ,1

| {z }

bones

,0,0, . . . ,0

| {z }

czeroes

) for somea,b,c ∈N.

Thus, this matrix is the transpose of a Petrie matrix. Hence, its determinant is ∈ {−1,0,1} (byGordon and Wilkinson 1974).

16 / 43

Expanding Petries in the Schur basis: the formula

Theorem: Let k be a positive integer. Letµ∈Par. Then, G(k)·sµ= X

λ∈Par

pet_k(λ, µ)sλ. Thus, for each m∈N, we have

G(k,m)·sµ= X

λ∈Par_m+|µ|

pet_k(λ, µ)sλ.

One proof of the Theorem uses alternants; the other uses the

“semi-skew Cauchy identity” X

λ∈Par

s_λ(x)s_λ/µ(y) =sµ(x)·

∞

Y

i,j=1

(1−x_iy_j)⁻¹

=s_µ(x)· X

λ∈Par

h_λ(x)m_λ(y) (for any µ∈Par and for two sets of indeterminates x = (x1,x2,x3, . . .) and y = (y1,y2,y3, . . .)).

17 / 43

Expanding Petries in the Schur basis: the formula

Theorem: Let k be a positive integer. Letµ∈Par. Then, G(k)·sµ= X

λ∈Par

pet_k(λ, µ)sλ. Thus, for each m∈N, we have

G(k,m)·sµ= X

λ∈Par_m+|µ|

pet_k(λ, µ)sλ.

One proof of the Theorem uses alternants; the other uses the

“semi-skew Cauchy identity” X

λ∈Par

s_λ(x)s_λ/µ(y) =sµ(x)·

∞

Y

i,j=1

(1−x_iy_j)⁻¹

=s_µ(x)· X

λ∈Par

h_λ(x)m_λ(y) (for any µ∈Par and for two sets of indeterminates x = (x1,x2,x3, . . .) and y = (y1,y2,y3, . . .)).

17 / 43

Expanding Petries in the Schur basis: the formula

Theorem: Let k be a positive integer. Letµ∈Par. Then, G(k)·sµ= X

λ∈Par

pet_k(λ, µ)sλ. Thus, for each m∈N, we have

G(k,m)·s_µ= X

λ∈Parm+|µ|

pet_k(λ, µ)s_λ. Corollary: Let k be a positive integer. Then,

G(k) = X

λ∈Par

pet_k(λ,∅)s_λ. Thus, for each m∈N, we have

G(k,m) = X

λ∈Par_m

pet_k(λ,∅)s_λ.

One proof of the Theorem uses alternants; the other uses the

“semi-skew Cauchy identity”

X

λ∈Par

s_λ(x)s_λ/µ(y) =s_µ(x)·

∞

Y

i,j=1

(1−x_iy_j)⁻¹

=s_µ(x)· X

λ∈Par

h_λ(x)m_λ(y) (for any µ∈Par and for two sets of indeterminates x = (x₁,x₂,x₃, . . .) and y = (y₁,y₂,y₃, . . .)).

17 / 43

Expanding Petries in the Schur basis: the formula

Theorem: Let k be a positive integer. Letµ∈Par. Then, G(k)·sµ= X

λ∈Par

pet_k(λ, µ)sλ. Thus, for each m∈N, we have

G(k,m)·sµ= X

λ∈Par_m+|µ|

pet_k(λ, µ)sλ.

One proof of the Theorem uses alternants; the other uses the

“semi-skew Cauchy identity”

X

λ∈Par

s_λ(x)s_λ/µ(y) =s_µ(x)·

∞

Y

i,j=1

(1−x_iy_j)⁻¹

=s_µ(x)· X

λ∈Par

h_λ(x)m_λ(y) (for any µ∈Par and for two sets of indeterminates x = (x1,x2,x3, . . .) and y = (y1,y2,y3, . . .)).

17 / 43

What are the Petrie numbers?

We have shown that pet_k(λ, µ)∈ {0,1,−1}, but what exactly is it?

Gordon and Wilkinson 1974 prove that Petrie matrices have determinants ∈ {0,1,−1}by induction. This is little help to us.

18 / 43

What are the Petrie numbers?

We have shown that pet_k(λ, µ)∈ {0,1,−1}, but what exactly is it?

Gordon and Wilkinson 1974prove that Petrie matrices have determinants ∈ {0,1,−1}by induction. This is little help to us.

18 / 43

What are the Petrie numbers? The easy case

Proposition: Letλ∈Par and k >0 be such that λ₁ ≥k. Then, pet_k(λ,∅) = 0.

To get a description in all other cases, recall the definition of transpose (aka conjugate) partitions:

Given a partition λ∈Par, we define the transpose partitionλ^t of λto be the partitionµ given by

µ_i =|{j ∈ {1,2,3, . . .} | λ_j ≥i}| for all i ≥1.

In terms of Young diagrams, this is just flipping the diagram of λacross the diagonal.

19 / 43

What are the Petrie numbers? The easy case

Proposition: Letλ∈Par and k >0 be such that λ₁ ≥k. Then, pet_k(λ,∅) = 0.

To get a description in all other cases, recall the definition of transpose (aka conjugate) partitions:

Given a partition λ∈Par, we define the transpose partitionλ^t of λto be the partitionµ given by

µ_i =|{j ∈ {1,2,3, . . .} | λ_j ≥i}| for all i ≥1.

In terms of Young diagrams, this is just flipping the diagram of λacross the diagonal.

19 / 43

What are the Petrie numbers? Formula for pet_k(λ,∅) Theorem: Let λ∈Par and k >0 be such that λ₁<k. Let µ=λ^t (the transpose partition of λ). Thus, µ_k = 0.

For eachi ∈ {1,2, . . . ,k−1}, set

β_i =µ_i−i and γ_i = 1 + (β_i −1) %k

| {z }

remainder ofβi−1 modulok

.

(a)If thek−1 numbersγ1, γ2, . . . , γk−1 are not distinct, then pet_k(λ,∅) = 0.

(b) If thek−1 numbersγ₁, γ₂, . . . , γk−1 are distinct, then pet_k(λ,∅) = (−1)^{(β1+β2+···+βk−1)+g+(γ1+γ2+···+γk−1)}, where

g = n

(i,j)∈ {1,2, . . . ,k−1}² | i <j andγ_i < γ_jo . Question: Is there such a description for pet_k(λ, µ) ?

20 / 43

What are the Petrie numbers? Formula for pet_k(λ,∅) Theorem: Let λ∈Par and k >0 be such that λ₁<k. Let µ=λ^t (the transpose partition of λ). Thus, µ_k = 0.

For eachi ∈ {1,2, . . . ,k−1}, set

β_i =µ_i−i and γ_i = 1 + (β_i −1) %k

| {z }

remainder ofβi−1 modulok

.

(a)If thek−1 numbersγ1, γ2, . . . , γk−1 are not distinct, then pet_k(λ,∅) = 0.

(b) If thek−1 numbersγ₁, γ₂, . . . , γk−1 are distinct, then pet_k(λ,∅) = (−1)^{(β1+β2+···+βk−1)+g+(γ1+γ2+···+γk−1)}, where

g = n

(i,j)∈ {1,2, . . . ,k−1}² | i <j andγ_i < γ_jo . Question: Is there such a description for pet_k(λ, µ) ?

20 / 43

Other properties

For anyk >0, we define a map f_k : Λ→Λ by setting fk(a) =a

x₁^k,x₂^k,x₃^k, . . .

for eacha∈Λ.

This map f_k is called the k-th Frobenius endomorphismof Λ.

(Also known as plethysm byp_k. Perhaps the nicest plethysm!) Theorem: Let k be a positive integer. Letm∈N. Then,

G(k,m) =X

i∈N

(−1)ⁱhm−ki ·f_k(e_i).

Theorem: Fix a positive integerk. Assume that 1−k is invertible in k. Then, the family

(G(k,m))_m≥1 = (G(k,1),G(k,2),G(k,3), . . .) is an algebraically independent generating set of the commutative k-algebra Λ.

Thus, products of several elements of this family form a basis of Λ (if 1−k is invertible in k). These bases remain to be studied.

21 / 43

Other properties

For anyk >0, we define a map f_k : Λ→Λ by setting fk(a) =a

x₁^k,x₂^k,x₃^k, . . .

for eacha∈Λ.

This map f_k is called the k-th Frobenius endomorphismof Λ.

(Also known as plethysm byp_k. Perhaps the nicest plethysm!) Theorem: Let k be a positive integer. Letm∈N. Then,

G(k,m) =X

i∈N

(−1)ⁱhm−ki ·f_k(e_i).

Theorem: Fix a positive integerk. Assume that 1−k is invertible in k. Then, the family

(G(k,m))_m≥1 = (G(k,1),G(k,2),G(k,3), . . .) is an algebraically independent generating set of the commutative k-algebra Λ.

Thus, products of several elements of this family form a basis of Λ (if 1−k is invertible in k). These bases remain to be studied.

21 / 43

Other properties

For anyk >0, we define a map f_k : Λ→Λ by setting fk(a) =a

x₁^k,x₂^k,x₃^k, . . .

for eacha∈Λ.

This map f_k is called the k-th Frobenius endomorphismof Λ.

(Also known as plethysm byp_k. Perhaps the nicest plethysm!) Theorem: Let k be a positive integer. Letm∈N. Then,

G(k,m) =X

i∈N

(−1)ⁱhm−ki ·f_k(e_i).

Theorem: Fix a positive integerk. Assume that 1−k is invertible in k. Then, the family

(G(k,m))_m≥1 = (G(k,1),G(k,2),G(k,3), . . .) is an algebraically independent generating set of the commutative k-algebra Λ.

Thus, products of several elements of this family form a basis of Λ (if 1−k is invertible in k). These bases remain to be studied.

21 / 43

The Liu–Polo conjecture

This all begin with the following conjecture (Liu and Polo, arXiv:1908.08432):

X

λ∈Par2n−1; (n−1,n−1,1).λ

mλ =

n−2

X

i=0

(−1)ⁱs(n−1,n−1−i,1ⁱ⁺¹) for anyn >1.

Here, the symbol. stands fordominance of partitions (also known as majorization); i.e., for two partitions λandµ, we have

λ . µ if and only if

(λ₁+λ₂+· · ·+λ_i ≥µ₁+µ₂+· · ·+µ_i for all i). Let me briefly outline how this conjecture can be proved.

22 / 43

The Liu–Polo conjecture, proof: 1

The partitions λ∈Par2n−1 satisfying (n−1,n−1,1). λ are precisely the partitionsλ∈Par2n−1 satisfyingλ_i <n for all i.

Thus,

X

λ∈Par2n−1; (n−1,n−1,1).λ

mλ=G(n,2n−1).

So it remains to show that G(n,2n−1) =

n−2

X

i=0

(−1)ⁱs(n−1,n−1−i,1ⁱ⁺¹).

The formula for pet_k(λ,∅) should be useful here, but the combinatorics is tortuous.

Instead, we can work algebraically:

23 / 43

The Liu–Polo conjecture, proof: 1

The partitions λ∈Par2n−1 satisfying (n−1,n−1,1). λ are precisely the partitionsλ∈Par2n−1 satisfyingλ_i <n for all i.

Thus,

X

λ∈Par2n−1; (n−1,n−1,1).λ

mλ=G(n,2n−1).

So it remains to show that G(n,2n−1) =

n−2

X

i=0

(−1)ⁱs(n−1,n−1−i,1ⁱ⁺¹).

The formula for pet_k(λ,∅) should be useful here, but the combinatorics is tortuous.

Instead, we can work algebraically:

23 / 43

The Liu–Polo conjecture, proof: 1

The partitions λ∈Par2n−1 satisfying (n−1,n−1,1). λ are precisely the partitionsλ∈Par2n−1 satisfyingλ_i <n for all i.

Thus,

X

λ∈Par2n−1; (n−1,n−1,1).λ

mλ=G(n,2n−1).

So it remains to show that G(n,2n−1) =

n−2

X

i=0

(−1)ⁱs(n−1,n−1−i,1ⁱ⁺¹).

The formula for pet_k(λ,∅) should be useful here, but the combinatorics is tortuous.

Instead, we can work algebraically:

23 / 43