The story of the symmetric group - Chennai Mathematical Institute

The story of thesymmetric group

Vipul Naik

A briefintroduction to thesymmetric group

The set of allpermutations

The group of allpermutations

Cycle type of apermutation

The concept of cycledecomposition

Cycle types andconjugacy classes

Two canonical maps

Blockconcatenations

General idea of ablock concatenationmap

For permutations

Block concatenationon set-partitions

Block concatenationon integer partitions

Cardinalitycomputations

Centralizers ofpermutations

Classifying partitions,hence set-partitionsand permutations

Concatenation-invariantstructures

Conjugation-invariantstructure

Concatenation-invariance

The leaderrepresentation

The Eulerian numbers

Young tableaux

So what’s next?

The story of the symmetric group

Vipul Naik

March 20, 2007


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline

A brief introduction to the symmetric groupThe set of all permutationsThe group of all permutations

Cycle type of a permutationThe concept of cycle decompositionCycle types and conjugacy classesTwo canonical maps

Block concatenationsGeneral idea of a block concatenation mapFor permutationsBlock concatenation on set-partitionsBlock concatenation on integer partitions

Cardinality computationsCentralizers of permutationsClassifying partitions, hence set-partitions and permutations

Concatenation-invariant structuresConjugation-invariant structureConcatenation-invarianceThe leader representationThe Eulerian numbers

Young tableauxSo what’s next?


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

What is a permutation?

Let S be a set. A permutation(defined) on S is a bijectivemapping S → S .

Equivalently a permutation is a function from S to S forwhich we can find an inverse function.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

What is a permutation?

Let S be a set. A permutation(defined) on S is a bijectivemapping S → S .Equivalently a permutation is a function from S to S forwhich we can find an inverse function.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

How do we describe a permutation?

We can describe a permutation σ of a set S by providing arule that computes σ(a) for each a ∈ S .

In particular, when S is a finite set, we can describe σ usingthe following two-line notation:

I The upper line lists the elements of S

I The lower line lists, under each element a ∈ S , thevalue σ(a).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

How do we describe a permutation?

We can describe a permutation σ of a set S by providing arule that computes σ(a) for each a ∈ S .In particular, when S is a finite set, we can describe σ usingthe following two-line notation:

I The upper line lists the elements of S

I The lower line lists, under each element a ∈ S , thevalue σ(a).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

How many permutations are there?

For a set S of size n, there are exactly n! permutations.

To prove this, first observe that the number of permutationsof a set S is finite and is dependent only on the cardinalityof S (and not on any additional structure with which S maybe endowed).Further, observe that it is equal to the number of bijectionsbetween any two sets of the same cardinality. Call thisnumber f (n).Now, pick a ∈ S . There are n possibilities for σ(a).Whatever value we choose for σ(a), there are n − 1 possiblevalues for the images of the remaining elements under σ.Thus, for each choice of σ(a) there are f (n− 1) possibilities.Thus:

f (n) = nf (n − 1)

And we have f (n) = n!


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


For a set S of size n, there are exactly n! permutations.To prove this, first observe that the number of permutationsof a set S is finite and is dependent only on the cardinalityof S (and not on any additional structure with which S maybe endowed).

Further, observe that it is equal to the number of bijectionsbetween any two sets of the same cardinality. Call thisnumber f (n).Now, pick a ∈ S . There are n possibilities for σ(a).Whatever value we choose for σ(a), there are n − 1 possiblevalues for the images of the remaining elements under σ.Thus, for each choice of σ(a) there are f (n− 1) possibilities.Thus:

f (n) = nf (n − 1)



Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


For a set S of size n, there are exactly n! permutations.To prove this, first observe that the number of permutationsof a set S is finite and is dependent only on the cardinalityof S (and not on any additional structure with which S maybe endowed).Further, observe that it is equal to the number of bijectionsbetween any two sets of the same cardinality. Call thisnumber f (n).

Now, pick a ∈ S . There are n possibilities for σ(a).Whatever value we choose for σ(a), there are n − 1 possiblevalues for the images of the remaining elements under σ.Thus, for each choice of σ(a) there are f (n− 1) possibilities.Thus:

f (n) = nf (n − 1)



Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


For a set S of size n, there are exactly n! permutations.To prove this, first observe that the number of permutationsof a set S is finite and is dependent only on the cardinalityof S (and not on any additional structure with which S maybe endowed).Further, observe that it is equal to the number of bijectionsbetween any two sets of the same cardinality. Call thisnumber f (n).Now, pick a ∈ S . There are n possibilities for σ(a).Whatever value we choose for σ(a), there are n − 1 possiblevalues for the images of the remaining elements under σ.Thus, for each choice of σ(a) there are f (n− 1) possibilities.Thus:

f (n) = nf (n − 1)



Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Fix the set once and for all

Since the structure of the set of permutations is the same forany set of size n, let us for convenience take the set of size nas {1, 2, . . . , n}.

In this case, we can dispense with the two lines of thetwo-line notation for a permutation and just specify one line– the second line. The first line is understood to be{1, 2, . . . , n}.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Fix the set once and for all

Since the structure of the set of permutations is the same forany set of size n, let us for convenience take the set of size nas {1, 2, . . . , n}.In this case, we can dispense with the two lines of thetwo-line notation for a permutation and just specify one line– the second line. The first line is understood to be{1, 2, . . . , n}.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Composing as multiplication

Since permutations are functions, we can compose them asfunctions, and the composite of two permutations is apermutation.

Further, the composition operation is associative, hence wehave an associative structure on the set of all permutations.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Composing as multiplication

Since permutations are functions, we can compose them asfunctions, and the composite of two permutations is apermutation.Further, the composition operation is associative, hence wehave an associative structure on the set of all permutations.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Inverses

By definition, a permutation is a function which has aninverse function. Clearly, the inverse must itself be apermutation.

Thus, we have on the set of permutations two structures:

I An associative multiplication given by functioncomposition

I An identity map that is an identity for the associativemultiplication

I An inverse map that is an inverse for this associativemultiplication

The set of permutations is now a group.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Inverses

By definition, a permutation is a function which has aninverse function. Clearly, the inverse must itself be apermutation.Thus, we have on the set of permutations two structures:

I An associative multiplication given by functioncomposition

I An identity map that is an identity for the associativemultiplication

I An inverse map that is an inverse for this associativemultiplication

The set of permutations is now a group.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Some notation

For any set S , the group of all permutations of the set S isdenoted as Sym(S) and is termed the symmetricgroup(defined) on S .

The symmetric group on {1, 2, . . . , n} is denoted as Sn andis termed the symmetric group on n letters.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Some notation

For any set S , the group of all permutations of the set S isdenoted as Sym(S) and is termed the symmetricgroup(defined) on S .The symmetric group on {1, 2, . . . , n} is denoted as Sn andis termed the symmetric group on n letters.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline








Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

A directed graph associated with a function

Let S be a finite set and f : S → S be a function. Then, wecan associate to f a directed graph with vertex set S asfollows.

For u ∈ S , make an edge directed from u to f (u).The outdegree of every vertex is 1, and f is a permutation ifand only if the indegree of every vertex is exactly one.But a directed graph with every vertex having indegree andoutdegree 1, is simply a disjoint union of directed cycles.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Let S be a finite set and f : S → S be a function. Then, wecan associate to f a directed graph with vertex set S asfollows.For u ∈ S , make an edge directed from u to f (u).

The outdegree of every vertex is 1, and f is a permutation ifand only if the indegree of every vertex is exactly one.But a directed graph with every vertex having indegree andoutdegree 1, is simply a disjoint union of directed cycles.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Let S be a finite set and f : S → S be a function. Then, wecan associate to f a directed graph with vertex set S asfollows.For u ∈ S , make an edge directed from u to f (u).The outdegree of every vertex is 1, and f is a permutation ifand only if the indegree of every vertex is exactly one.

But a directed graph with every vertex having indegree andoutdegree 1, is simply a disjoint union of directed cycles.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Let S be a finite set and f : S → S be a function. Then, wecan associate to f a directed graph with vertex set S asfollows.For u ∈ S , make an edge directed from u to f (u).The outdegree of every vertex is 1, and f is a permutation ifand only if the indegree of every vertex is exactly one.But a directed graph with every vertex having indegree andoutdegree 1, is simply a disjoint union of directed cycles.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Cycle decomposition of a permutation

A cycle is a permutation such that the graph associated withit is a cyclic graph.

In other words, a cycle on S is a permutation where, for anya ∈ S , the set comprising {a, σ(a), σ2(a), . . . , } is the wholeof S .The graph of a permutation is a disjoint union of cycles.Hence any permutation is a product of pairwise disjointcycles (that is, cycles with no elements in common betweenany two).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


A cycle is a permutation such that the graph associated withit is a cyclic graph.In other words, a cycle on S is a permutation where, for anya ∈ S , the set comprising {a, σ(a), σ2(a), . . . , } is the wholeof S .

The graph of a permutation is a disjoint union of cycles.Hence any permutation is a product of pairwise disjointcycles (that is, cycles with no elements in common betweenany two).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


A cycle is a permutation such that the graph associated withit is a cyclic graph.In other words, a cycle on S is a permutation where, for anya ∈ S , the set comprising {a, σ(a), σ2(a), . . . , } is the wholeof S .The graph of a permutation is a disjoint union of cycles.

Hence any permutation is a product of pairwise disjointcycles (that is, cycles with no elements in common betweenany two).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


A cycle is a permutation such that the graph associated withit is a cyclic graph.In other words, a cycle on S is a permutation where, for anya ∈ S , the set comprising {a, σ(a), σ2(a), . . . , } is the wholeof S .The graph of a permutation is a disjoint union of cycles.Hence any permutation is a product of pairwise disjointcycles (that is, cycles with no elements in common betweenany two).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Cycle type of a permutation

The cycle type of a permutation is the following data thatcan be inferred from the cycle decomposition. It is asequence (i1, i2, . . .) where ij denotes the number of cycles oflength j in the cycle decomposition of the given permutation.

The relation of having the same cycle type is an equivalencerelation that partitions the set of all permutations intoequivalence classes.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Cycle type of a permutation

The cycle type of a permutation is the following data thatcan be inferred from the cycle decomposition. It is asequence (i1, i2, . . .) where ij denotes the number of cycles oflength j in the cycle decomposition of the given permutation.The relation of having the same cycle type is an equivalencerelation that partitions the set of all permutations intoequivalence classes.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

The number of cycle types

Any cycle type (i1, i2, . . .) for permutations on a set of size nmust satisfy the following:∑

k

kik = n

In other words, cycle types correspond to unorderedpartitions of n. Thus, the number of cycle types forpermutations of length n is p(n), the number of partitions ofthe integer n.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Conjugation as relabelling

In any group, we have certain inner automorphism definedvia conjugation by an element. For g ∈ G , the conjugationmap by g is cg : x 7→ gxg−1.

When G = Sn, the conjugation map by g ∈ G has aparticularly nice description. Namely cg (x) is the map thatsends g(a) to g(x(a)).In other words, cg (x) is x twisted via a g -relabelling.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Conjugation as relabelling

In any group, we have certain inner automorphism definedvia conjugation by an element. For g ∈ G , the conjugationmap by g is cg : x 7→ gxg−1.When G = Sn, the conjugation map by g ∈ G has aparticularly nice description. Namely cg (x) is the map thatsends g(a) to g(x(a)).In other words, cg (x) is x twisted via a g -relabelling.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Conjugation preserves cycle type

Since conjugation only relabels the elements, it does notchange the qualitative characteristics of the permutation.Thus, it sends each permutation to another permutationwith the same cycle type.

Conversely, given any two permutations of the same cycletype, there is a conjugation that takes one to the other.Hence two permutations have the same cycle type if andonly if they are conjugate.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Since conjugation only relabels the elements, it does notchange the qualitative characteristics of the permutation.Thus, it sends each permutation to another permutationwith the same cycle type.Conversely, given any two permutations of the same cycletype, there is a conjugation that takes one to the other.

Hence two permutations have the same cycle type if andonly if they are conjugate.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Since conjugation only relabels the elements, it does notchange the qualitative characteristics of the permutation.Thus, it sends each permutation to another permutationwith the same cycle type.Conversely, given any two permutations of the same cycletype, there is a conjugation that takes one to the other.Hence two permutations have the same cycle type if andonly if they are conjugate.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Centralizers and conjugacy classes

We now have a somewhat group-theoretic twist to theproblem of computing the number of permutations of agiven cycle type.

The permutations of cycle type i = (i1, i2, . . .) form an orbitunder the group’s action on itself by conjugation.Thus, the number of permutations is the cardinality of thegroup divided by the cardinality of the isotropy group of anyone permutation having that cycle type.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


We now have a somewhat group-theoretic twist to theproblem of computing the number of permutations of agiven cycle type.The permutations of cycle type i = (i1, i2, . . .) form an orbitunder the group’s action on itself by conjugation.

Thus, the number of permutations is the cardinality of thegroup divided by the cardinality of the isotropy group of anyone permutation having that cycle type.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


We now have a somewhat group-theoretic twist to theproblem of computing the number of permutations of agiven cycle type.The permutations of cycle type i = (i1, i2, . . .) form an orbitunder the group’s action on itself by conjugation.Thus, the number of permutations is the cardinality of thegroup divided by the cardinality of the isotropy group of anyone permutation having that cycle type.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Combinatorial structures of interest to us

Five combinatorial structures of interest to us are:

I Sn or Sym(n) is the set of all permutations on a set oforder n. The cardinality of this set is n!

I OB(n) is the set of all partitions of a set of n elementsinto an ordered sequence of sets.

I B(n) is the set of all equivalence relations (or setpartitions) of a set of n elements. The cardinality ofthis set is Bn (the Bell number)

I OP(n) is the set of all ordered partitions of n intononnegative integers

I P(n) is the set of all unordered partitions of n intononnegative integers. The cardinality of this set is p(n)


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Canonical map from permutations toset-partitions

There is a map:

Pr : Sn → B(n)

that sends a permutation σ to the equivalence relation ofbeing in the same cycle in the cycle decomposition of σ.

Another way of putting this is that it gives the equivalencerelation of being in the same connected component of thedirected graph associated with σ.Note that this map is independent of the labels on theelements of the set, and thus, in particular, it is equivariantunder inner automorphisms.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Canonical map from permutations toset-partitions

There is a map:

Pr : Sn → B(n)

that sends a permutation σ to the equivalence relation ofbeing in the same cycle in the cycle decomposition of σ.Another way of putting this is that it gives the equivalencerelation of being in the same connected component of thedirected graph associated with σ.Note that this map is independent of the labels on theelements of the set, and thus, in particular, it is equivariantunder inner automorphisms.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Canonical map from set-partitions to unorderedinteger partitions

Given a set partition, the corresponding unordered integerpartition is the partition that uses each integer j as manytimes as there are subsets of size j .

This gives a map:

Upr : B(n) → P(n)

The composite CT = Upr ◦ Pr is the cycle type map thatsends each permutation to its cycle type.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Canonical map from set-partitions to unorderedinteger partitions

Given a set partition, the corresponding unordered integerpartition is the partition that uses each integer j as manytimes as there are subsets of size j .This gives a map:

Upr : B(n) → P(n)

The composite CT = Upr ◦ Pr is the cycle type map thatsends each permutation to its cycle type.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Cardinality questions that we are interested in

We have the following maps:

Sn → B(n) → P(n)

Two questions of interest:

I Given π ∈ P(n), what can we say about the sizes ofUpr−1(π) and Pr−1 ◦ Upr−1(π)?

I Suppose we construct a further map:

P(n) → {1, 2, . . . , n}

What can we say about the inverse images ofk ∈ {1, 2, . . . , n} via the composite mappings?


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline








Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

General notion of block concatenation

Suppose we have a family of sets Tm for every integerm ∈ N. Then, a block concatenation structure on the Tms isthe following kind of structure:For each m, n ∈ N, we have a map:

Φm,n : Tm × Tn → Tm+n

Satisfying a certain associativity condition.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Block concatenation of more than two things

Given an ordered partition n = m1 + m2 + . . . + mr , we havean injective homomorphism:

Φm1,m2,...,mr : Tm1 × Tm2 × . . .× Tmr → Tn

This essentially keeps applying the block concatenation twoat a time, and uses associativity to guarantee that the resultis well-defined.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Block concatenation on permutations

Let m, n ∈ N, with g ∈ Sm and h ∈ Sn. Then, consider theelement of Sm+n defined as follows: It acts like g on the firstm letters and acts like h on the remaining n letters (treatingm + k as k).

This gives an injective homomorphism (which is a blockconcatenation):

Φm,n : Sm × Sn → Sm+n

The image of Φm,n is termed the Young subgroup for theordered pair (m, n).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Block concatenation on permutations

Let m, n ∈ N, with g ∈ Sm and h ∈ Sn. Then, consider theelement of Sm+n defined as follows: It acts like g on the firstm letters and acts like h on the remaining n letters (treatingm + k as k).This gives an injective homomorphism (which is a blockconcatenation):

Φm,n : Sm × Sn → Sm+n

The image of Φm,n is termed the Young subgroup for theordered pair (m, n).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Block concatenation of multiple permutations

Given an ordered integer partition n = m1 + m2 + . . . + mr ,we have a map:

Φm1,m2,...,mr : Sm1 × Sm2 × . . .× Smr → Sn

The image of this map is termed the Young subgroupcorresponding to the ordered partition.

The Young subgroup can equivalently be thought of as thoseelements of the group of all permutations that stabilize eachpart in the partition.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Block concatenation of multiple permutations

Given an ordered integer partition n = m1 + m2 + . . . + mr ,we have a map:

Φm1,m2,...,mr : Sm1 × Sm2 × . . .× Smr → Sn

The image of this map is termed the Young subgroupcorresponding to the ordered partition.The Young subgroup can equivalently be thought of as thoseelements of the group of all permutations that stabilize eachpart in the partition.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

For ordered set-partitions

Let n = m1 + m2 + . . . + mr be an ordered integer partitionof n. Then, we have an injective map:

Φm1,m2,...,mr : OB(m1)× OB(m2)× . . .× OB(mr ) → OB(n)

that simply takes the union of the ordered set partitions forthe subsets of sizes mi for each i .


Vipul Naik







Two canonical maps

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Young tableaux

So what’s next?

For unordered set-partitions

We can also perform a block concatenation on unordered setpartitions. This will again give an injective map.

(The mapcontinues to remain injective because, given any partitionthat arises as the image of a block concatenation, we canuniquely retrieve the set-partitions in each block that gaverise to it).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

For unordered set-partitions

We can also perform a block concatenation on unordered setpartitions. This will again give an injective map. (The mapcontinues to remain injective because, given any partitionthat arises as the image of a block concatenation, we canuniquely retrieve the set-partitions in each block that gaverise to it).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

For ordered integer partitions

Given an ordered integer partition n = m1 + m2 + . . .mr , wehave a map:

Φm1,m2,...,mr : OP(m1)×OP(m2)× . . .×OP(m−r) → OP(n)

Which simply concatenates the partitions one after the other.

This map is again injective.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

For ordered integer partitions

Given an ordered integer partition n = m1 + m2 + . . .mr , wehave a map:

Φm1,m2,...,mr : OP(m1)×OP(m2)× . . .×OP(m−r) → OP(n)

Which simply concatenates the partitions one after the other.This map is again injective.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

For unordered integer partitions

Given an unordered integer partition n = m1 + m2 . . .mr , wehave a map:

Φm1,m2,...,mr : P(m1)× P(m2)× . . .× P(mr ) → P(n)

Unlike all the previous maps this one is not injective. Thatis, there could be different unordered integer partition tuplesthat give rise to the same unordered integer partition underblock concatenation.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline








Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Stabilizer for a whole partition

Given an unordered set partition of the set {1, 2, . . . , n} wecan compute the set of those permutations that fix every setin the partition. This, it turns out, is isomorphic to theproduct of symmetric groups for every part.

We also have a weaker notion of symmetry for a partition.Namely, we are interested in those permutations thatpreserve the equivalence relation of the partition. In otherwords, we are interested in permutations that send each setof the partition completely to another set of the partition.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Stabilizer for a whole partition

Given an unordered set partition of the set {1, 2, . . . , n} wecan compute the set of those permutations that fix every setin the partition. This, it turns out, is isomorphic to theproduct of symmetric groups for every part.We also have a weaker notion of symmetry for a partition.Namely, we are interested in those permutations thatpreserve the equivalence relation of the partition. In otherwords, we are interested in permutations that send each setof the partition completely to another set of the partition.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Cardinality of the symmetry group of a partition

Consider an unordered partition of n with ij occurrences of jfor each j .

Then the group of symmetries that preserve the equivalencerelation induced by the partition is described as follows. It isa direct product of groups D(j , ij) where D(j , ij) is the groupof permutations on the union of the blocks of size j thattake each block to a block.It turns out that:

D(j , ij) = (Sj)ij o Sij

And the symmetry group for a partition is:∏j

D(j , ij)


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Consider an unordered partition of n with ij occurrences of jfor each j .Then the group of symmetries that preserve the equivalencerelation induced by the partition is described as follows. It isa direct product of groups D(j , ij) where D(j , ij) is the groupof permutations on the union of the blocks of size j thattake each block to a block.

It turns out that:



D(j , ij)


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Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


Consider an unordered partition of n with ij occurrences of jfor each j .Then the group of symmetries that preserve the equivalencerelation induced by the partition is described as follows. It isa direct product of groups D(j , ij) where D(j , ij) is the groupof permutations on the union of the blocks of size j thattake each block to a block.It turns out that:



D(j , ij)


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Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Description of the centralizer

let σ be a permutation of cycle type (i1, i2, . . .)Then:

CSn(σ) =∏j

C (j , ij)

where:

C (j , ij) = (Z/jZ)ij o Sij

this is a subgroup of the group of all symmetries of thepartition.


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Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Some explicit formulae

Let π = (i1, i2, . . .) be an unordered partition of n. Then, wehave:

I The number of set-partitions corresponding to π is[Sn : D(j , ij)]. Recall that this is CT−1(π).

I The number of permutations corresponding to π is[Sn : C (j , ij)]. Recall that this is Pr−1(π).


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Two canonical maps

Blockconcatenations


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Young tableaux

So what’s next?

Recall of motivations

We had maps:

Sn → B(n) → P(n)

We had observed that given any map k : P(n) → S thatessentially “classifies” unordered integer partitions, we getcorresponding maps Sn → S and B(n) → S . S here mayusually be {1, 2, . . . , n} or {0, 1, . . . , n}.

This essentially divides Sn or B(n) into n or n + 1 parts.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Recall of motivations

We had maps:

Sn → B(n) → P(n)

We had observed that given any map k : P(n) → S thatessentially “classifies” unordered integer partitions, we getcorresponding maps Sn → S and B(n) → S . S here mayusually be {1, 2, . . . , n} or {0, 1, . . . , n}.This essentially divides Sn or B(n) into n or n + 1 parts.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

The total number of partsOne possibility for k is the total number of partscorresponding to the unordered integer partition, viz for anunordered integer partition (i!, i2, . . .), we define:

k =∑

j

ij

We can partition the set of partitions based on the value ofk (k can range from 1 to n).Corresponding to this, we get the following way of classifyingpermutations:

I Permutations get classified according to the number ofcycles. The number of permutations with k cycles istermed s ′(n, k) and is called the unsigned Stirlingnumber of the first kind.

I Set partitions get classified according to the number ofsubsets. The number of set-partitions with k subsets istermed S(n, k) and is called the Stirling number of thesecond kind.


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Blockconcatenations


For permutations











Young tableaux

So what’s next?

The number of singleton parts

Another approach is to define k = i1, that is, we classifyunordered integer partitions of n according to the number oftimes 1 occurs in the integer partition.

As per this classification:

I The number of permutations with k fixed points is(nk

)Dn−k where Dr is the number of permutations on r

elements with no fixed point.

I The number of unordered set partitions with k fixedpoints is

(nk

)B2(n − k) where B2(r) is the number of

unordered set partitions with every set having size atleast 2.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

The number of singleton parts

Another approach is to define k = i1, that is, we classifyunordered integer partitions of n according to the number oftimes 1 occurs in the integer partition.As per this classification:

I The number of permutations with k fixed points is(nk

)Dn−k where Dr is the number of permutations on r

elements with no fixed point.

I The number of unordered set partitions with k fixedpoints is

(nk

)B2(n − k) where B2(r) is the number of

unordered set partitions with every set having size atleast 2.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline








Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

What we have seen so far

So far, the various structural elements we have seen areunchanged upto relabelling. In other words, if we altered thelabels, the maps would also simply get relabelled. The mapsdo not fundamentally depend on the order of the elements.

One way of looking at the above statement is as follows:For any set S of size n, we can define:

Sym(S) → B(S) → P(n)

Now, Sym(S) acts on S , and hence also on B(S) and onSym(S) (by inner automorphism). The claim is that themap from Sym(S) to B(S) is equivariant under Sym(S)action, and the right map is invariant under Sym(S) action.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

What we have seen so far

So far, the various structural elements we have seen areunchanged upto relabelling. In other words, if we altered thelabels, the maps would also simply get relabelled. The mapsdo not fundamentally depend on the order of the elements.One way of looking at the above statement is as follows:For any set S of size n, we can define:

Sym(S) → B(S) → P(n)

Now, Sym(S) acts on S , and hence also on B(S) and onSym(S) (by inner automorphism). The claim is that themap from Sym(S) to B(S) is equivariant under Sym(S)action, and the right map is invariant under Sym(S) action.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Why this is restrictive

Looking only at structures that are invariant underconjugation or relabelling causes us to miss out on a lot offun. For instance, we cannot try to see whether 1 occursbefore 2, whether 2 occurs before 3, how the numbers riseand fall etc. Thus, we’d ideally like to look for structuresand maps that can exploit the total ordering of {1, 2, . . . , n}.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Concatenation-invariance: what it means

Suppose we have two sequences An and Dn of sets, bothequipped with block concatenation maps, and a mapfn : An → Dn for each n.We say that fns are compatible with the concatenation (orconcatenation-invariant) if concatenating before applying thefns has the same effect as concatenating after applying thefns.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Why we would like concatenation-invariance

The broad reason is that we want that if something (apermutation, a set-partition, or whatever) does not mix upthe first m elements with the last n elements, its behaviouron the first m elements should be like something for just melements and its behaviour for the last n elements should bejust like for n elements.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Some motivation

Recall that we have a natural map:

Upr : Sn → B(n)

Which takes a permutation and outputs the set partitioninduced by its cycle decomposition.Intrinsically, this set partition is unordered. However, if wethink of the underlying set as {1, 2, . . . , n}, we can order theparts by their least elements. That is, for disjoint subsets Aand B of {1, 2, . . . , n} say that A is less than B if the leastelement of A is less than the least element of B.

This gives a map:

M1 : Sn → OB(n)

And incidentally, also a map to OP(n).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Some motivation

Recall that we have a natural map:

Upr : Sn → B(n)

Which takes a permutation and outputs the set partitioninduced by its cycle decomposition.Intrinsically, this set partition is unordered. However, if wethink of the underlying set as {1, 2, . . . , n}, we can order theparts by their least elements. That is, for disjoint subsets Aand B of {1, 2, . . . , n} say that A is less than B if the leastelement of A is less than the least element of B.This gives a map:

M1 : Sn → OB(n)

And incidentally, also a map to OP(n).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Another map

We now consider another map:

M2 : Sn → OB(n)

And thus, also a map to OP(n).That takes a given permutation, writes it in one-linenotation, and then breaks the permutation at eachleft-to-right minimum.

It now turns out that we can define a bijection B : Sn → Sn

such that B ◦M1 = M2.In other words, we can take a permutation and write anotherpermutation such that the one-line notation of one interpretsthe cycle decomposition of the other.This map is concatenation-invariant.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Another map


M2 : Sn → OB(n)

And thus, also a map to OP(n).That takes a given permutation, writes it in one-linenotation, and then breaks the permutation at eachleft-to-right minimum.It now turns out that we can define a bijection B : Sn → Sn

such that B ◦M1 = M2.In other words, we can take a permutation and write anotherpermutation such that the one-line notation of one interpretsthe cycle decomposition of the other.

This map is concatenation-invariant.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Another map


M2 : Sn → OB(n)

And thus, also a map to OP(n).That takes a given permutation, writes it in one-linenotation, and then breaks the permutation at eachleft-to-right minimum.It now turns out that we can define a bijection B : Sn → Sn

such that B ◦M1 = M2.In other words, we can take a permutation and write anotherpermutation such that the one-line notation of one interpretsthe cycle decomposition of the other.This map is concatenation-invariant.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Yet another map!The map described above takes a permutation and breaks itat the left-to-right cumulative minima. Another interestingmap is one where we try to see whether the value is rising orfalling in the immediate sense.

We define a map:

A : Sn → OB(n)

That sends a permutation to the ordered set partitions thatbreak the permutations at the points where it falls. Thus,each part in the ordered partition is a rising run in thepermutation.This also gives a map:

A′ : Sn → OP(n)

And another:

A′′ : Sn → P(n)


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Yet another map!The map described above takes a permutation and breaks itat the left-to-right cumulative minima. Another interestingmap is one where we try to see whether the value is rising orfalling in the immediate sense.We define a map:

A : Sn → OB(n)

That sends a permutation to the ordered set partitions thatbreak the permutations at the points where it falls. Thus,each part in the ordered partition is a rising run in thepermutation.This also gives a map:

A′ : Sn → OP(n)

And another:

A′′ : Sn → P(n)


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


We have a map:

A′′ : Sn → P(n)

which counts the number ij of rising runs in a givepermutation of every length j .

The sum∑

j ij gives the total number of rising runs.We thus have a map:

A′′′ : Sn → {1, 2, . . . , n}

The size of the inverse image of a given k is termed theEulerian number A(n, k).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


We have a map:

A′′ : Sn → P(n)

which counts the number ij of rising runs in a givepermutation of every length j .The sum

∑j ij gives the total number of rising runs.

We thus have a map:

A′′′ : Sn → {1, 2, . . . , n}

The size of the inverse image of a given k is termed theEulerian number A(n, k).


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

Outline








Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

A close look at what’s been so far

So far, we have looked at various maps from Sn to sets suchas B(n), P(n) and {1, 2, . . . , n} (often, with one mapfactoring through the other).

The maps we considered initially depended only on the cycledecomposition: a conjugation-invariant structure. The mapsinvolving left-to-right minima and rises and falls used theorder structure.However, even the maps that have so far used the order,have used it very ineffectively. They haven’t exploited muchof the information inherent in a permutation.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


So far, we have looked at various maps from Sn to sets suchas B(n), P(n) and {1, 2, . . . , n} (often, with one mapfactoring through the other).The maps we considered initially depended only on the cycledecomposition: a conjugation-invariant structure. The mapsinvolving left-to-right minima and rises and falls used theorder structure.

However, even the maps that have so far used the order,have used it very ineffectively. They haven’t exploited muchof the information inherent in a permutation.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


So far, we have looked at various maps from Sn to sets suchas B(n), P(n) and {1, 2, . . . , n} (often, with one mapfactoring through the other).The maps we considered initially depended only on the cycledecomposition: a conjugation-invariant structure. The mapsinvolving left-to-right minima and rises and falls used theorder structure.However, even the maps that have so far used the order,have used it very ineffectively. They haven’t exploited muchof the information inherent in a permutation.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?

A two-dimensional information storagemechanism

The left-to-right minimum approach was poor in terms ofinformation storage because it neglected all the things thatwere not left-to-right minima. For instance, for allpermutations with the first entry 1, the left-to-rightminimum information is the same.

The rising and falling runs approach was poor because itlooked only at the local relationships, and it failed to capturethe way entries far away linked with each other.Thus, if we want to preserve maximum information about apermutation, we have to store both the local behaviour (thatwith respect to the things just before) and a globalbehaviour (That with respect to all the things that havebeen encountered so far).The method for doing this is Young tableaux.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


The left-to-right minimum approach was poor in terms ofinformation storage because it neglected all the things thatwere not left-to-right minima. For instance, for allpermutations with the first entry 1, the left-to-rightminimum information is the same.The rising and falling runs approach was poor because itlooked only at the local relationships, and it failed to capturethe way entries far away linked with each other.

Thus, if we want to preserve maximum information about apermutation, we have to store both the local behaviour (thatwith respect to the things just before) and a globalbehaviour (That with respect to all the things that havebeen encountered so far).The method for doing this is Young tableaux.


Vipul Naik







Two canonical maps

Blockconcatenations


For permutations











Young tableaux

So what’s next?


The left-to-right minimum approach was poor in terms ofinformation storage because it neglected all the things thatwere not left-to-right minima. For instance, for allpermutations with the first entry 1, the left-to-rightminimum information is the same.The rising and falling runs approach was poor because itlooked only at the local relationships, and it failed to capturethe way entries far away linked with each other.Thus, if we want to preserve maximum information about apermutation, we have to store both the local behaviour (thatwith respect to the things just before) and a globalbehaviour (That with respect to all the things that havebeen encountered so far).The method for doing this is Young tableaux.

The story of the symmetric group - Chennai Mathematical Institute

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