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RSK pairs for two nonidentity involutions

Example

For $w = (2, 1, 4, 3)$ row insertion gives $P = \begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}, Q = \begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array},$ so $P = Q$ as required for the involution $w = w^{- 1}$ . For $w = (3, 2, 1)$ one gets $P = Q = \begin{array}{l} 1 \\ 2 \\ 3 \end{array}$ . The involution $(2, 1, 4, 3)$ is not the identity, so $P = Q$ is not equivalent to $P$ being a single row; for $n = 3$ the derivation $\sum_{λ ⊢ 3} f^{λ} = 1 + 2 + 1 = 4$ matches the four involutions $i d, (1 2), (1 3), (2 3)$ (in one-line form $123$ , $213$ , $321$ , $132$ ).

Facts & Assumptions

Given: The words $w = (2, 1, 4, 3)$ and $w = (3, 2, 1)$ and their RSK pairs, and the partitions of $3$ .

[L1]

A permutation is an involution exactly when its RSK pair satisfies $P = Q$ ; the map $w \mapsto P (w)$ is a bijection from the involutions of ${1, \dots, n}$ onto the standard tableaux with $n$ boxes (Involutions are counted by standard tableaux, RSK interchanges the insertion and recording tableaux under inversion).

[L2]

The RSK map is a bijection from the permutations of ${1, \dots, n}$ in one-line form onto the pairs of standard tableaux of common shape $λ ⊢ n$ (The Robinson-Schensted correspondence).

[F1]

The partitions of $3$ are $(3), (2, 1), (1, 1, 1)$ ; the standard tableaux with three boxes are $(1, 2, 3)$ of shape $(3)$ , the two tableaux $(1, 2; 3)$ and $(1, 3; 2)$ of shape $(2, 1)$ , and $(1; 2; 3)$ of shape $(1, 1, 1)$ , so $\sum_{λ ⊢ 3} f^{λ} = 1 + 2 + 1 = 4$ , and the number of involutions of ${1, 2, 3}$ equals this sum (Involutions are counted by standard tableaux).

Verification

technique · direct

1.1L2given

( $w = (2, 1, 4, 3)$ .) Inserting $2, 1$ gives the first column $(2)$ , then $(1, 2)$ after $1$ displaces $2$ ; inserting $4$ appends it at the end of the first row, giving $(1, 4; 2)$ ; inserting $3$ replaces $4$ in the first row and appends $4$ in the second row, giving $P = (1, 3; 2, 4)$ . The added boxes in order are $(1, 1), (2, 1), (1, 2), (2, 2)$ , so $Q$ carries $1, 2, 3, 4$ in those boxes, i.e. $Q = (1, 3; 2, 4) = P$ .

1.2L2given

( $w = (3, 2, 1)$ .) Inserting $3, 2, 1$ successively replaces the first row entry each time and appends downwards, giving the single column $P = (1; 2; 3)$ ; the added boxes are $(1, 1), (2, 1), (3, 1)$ , so $Q = (1; 2; 3) = P$ .

2.1L1step 1.1step 1.2given

(Consistency with the criterion.) Both words are involutions: $(2, 1, 4, 3) = (1 2) (3 4)$ and $(3, 2, 1) = (1 3)$ , and in both cases step 1.1 or step 1.2 found $P = Q$ , as [L1] requires; the shapes $(2, 2)$ and $(1, 1, 1)$ are different, so $P = Q$ does not force a single shape.

2.2L1step 1.1step 1.2given

(Not only single rows.) The identity $123$ has RSK pair $P = Q = (1, 2, 3)$ of one-row shape $(3)$ , while the involution $w = (2, 1, 4, 3)$ has $P = Q$ of shape $(2, 2)$ by step 1.1 and the involution $(3, 2, 1)$ has $P = Q$ of shape $(1, 1, 1)$ by step 1.2. These non-row examples show that $P = Q$ is not equivalent to $P$ being a single row.

3.1F1L1step 1.2algebra∎

( $n = 3$ count.) By [F1], $\sum_{λ ⊢ 3} f^{λ} = f^{(3)} + f^{(2, 1)} + f^{(1, 1, 1)} = 1 + 2 + 1 = 4$ ; the four involutions of ${1, 2, 3}$ are the identity $123$ , the transpositions $(1 2) = 213$ , $(1 3) = 321$ and $(2 3) = 132$ , also four in number, matching the bijection of [L1] in size $3$ .

Depends on

Used by

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Sources

Charlotte Chan, Representation Theory of Symmetric Groups (Oxford Hilary Term 2011 lecture notes, 40 PDF pages) (standard reference, not scraped)
Donald E. Knuth, Permutations, Matrices, and Generalized Young Tableaux, Pacific Journal of Mathematics 34 (1970), 709-727 (standard reference, not scraped)
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics (263 pp.) (standard reference, not scraped)