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The Story of the Symmetric Group The story of the symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all The story of the symmetric group permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and Vipul Naik conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation March 20, 2007 map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Equivalently a permutation is a function from S to S for which we can find an inverse function. The story of the What is a permutation? symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation Let S be a set. A permutation(defined) on S is a bijective The concept of cycle decomposition mapping S → S. Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the What is a permutation? symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation Let S be a set. A permutation(defined) on S is a bijective The concept of cycle decomposition mapping S → S. Cycle types and conjugacy classes Equivalently a permutation is a function from S to S for Two canonical maps which we can find an inverse function. Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? In particular, when S is a finite set, we can describe σ using the following two-line notation: I The upper line lists the elements of S I The lower line lists, under each element a ∈ S, the value σ(a). The story of the How do we describe a permutation? symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I The upper line lists the elements of S I The lower line lists, under each element a ∈ S, the value σ(a). The story of the How do we describe a permutation? symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation In particular, when S is a finite set, we can describe σ using The concept of cycle decomposition the following two-line notation: Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the How do we describe a permutation? symmetric group Vipul Naik A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation In particular, when S is a finite set, we can describe σ using The concept of cycle decomposition the following two-line notation: Cycle types and conjugacy classes Two canonical maps I The upper line lists the elements of S Block concatenations I The lower line lists, under each element a ∈ S, the General idea of a block concatenation value σ(a). map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? To prove this, first observe that the number of permutations of a set S is finite and is dependent only on the cardinality of S (and not on any additional structure with which S may be endowed). Further, observe that it is equal to the number of bijections between any two sets of the same cardinality. Call this number f (n). Now, pick a ∈ S. There are n possibilities for σ(a). Whatever value we choose for σ(a), there are n − 1 possible values 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! The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Further, observe that it is equal to the number of bijections between any two sets of the same cardinality. Call this number f (n). Now, pick a ∈ S. There are n possibilities for σ(a). Whatever value we choose for σ(a), there are n − 1 possible values 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! The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief To prove this, first observe that the number of permutations introduction to the symmetric group of a set S is finite and is dependent only on the cardinality The set of all permutations of S (and not on any additional structure with which S may The group of all be endowed). permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer
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