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A full cogood-addition path from the empty partition to (3,1,1) at p = 3

Example

At $p = 3$ the $p$ -restricted partition $(3, 1, 1)$ is reached from the empty partition by the five cogood additions $\emptyset \overset{0}{\to} (1) \overset{2}{\to} (1, 1) \overset{1}{\to} (1, 1, 1) \overset{1}{\to} (2, 1, 1) \overset{2}{\to} (3, 1, 1),$ that is, $(1) = {\tilde{f}}_{0} \emptyset$ , $(1, 1) = {\tilde{f}}_{2} (1)$ , $(1, 1, 1) = {\tilde{f}}_{1} (1, 1)$ , $(2, 1, 1) = {\tilde{f}}_{1} (1, 1, 1)$ , $(3, 1, 1) = {\tilde{f}}_{2} (2, 1, 1)$ , where the labels record the residues of the added nodes (Good-node deletion and cogood-node addition operators). Reversing the path gives the good deletions $(3, 1, 1) \overset{2}{\to} (2, 1, 1) \overset{1}{\to} (1, 1, 1) \overset{1}{\to} (1, 1) \overset{2}{\to} (1) \overset{0}{\to} \emptyset .$ The relevant reduced signatures are: $(1)$ at $i = 0$ : $(-)$ ; $(1, 1)$ at $i = 2$ : $(-)$ ; $(1, 1, 1)$ at $i = 1$ : $(+, -)$ ; $(2, 1, 1)$ at $i = 1$ : $(-, -)$ ; $(3, 1, 1)$ at $i = 2$ : $(-)$ ; and $(1, 1)$ at $i = 1$ : $(+, +)$ , $(2, 1, 1)$ at $i = 2$ : $(+)$ . The word $(-, -)$ for $(2, 1, 1)$ has two normal nodes of the same residue, while $(+, -)$ for $(1, 1, 1)$ shows that a reduced signature may contain both signs.

Facts & Assumptions

Given: The prime $p = 3$ , the residues of a node computed as $c - r m o d 3$ , and the partitions $\emptyset$ , $(1)$ , $(1, 1)$ , $(1, 1, 1)$ , $(2, 1, 1)$ , $(3, 1, 1)$ .

[F1]

The reduced $i$ -signature of a partition is obtained by listing the addable and removable $i$ -nodes top to bottom, marking them $+$ and $-$ , and cancelling $- +$ pairs; the good $i$ -node is the removable node of the first surviving $-$ , the cogood $i$ -node is the addable node of the last surviving $+$ , and ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ delete the good node and add the cogood node (Good-node deletion and cogood-node addition operators, Residue signatures, normal, good, conormal and cogood nodes).

[F2]

For $p$ -restricted $μ, ν$ one has $μ = {\tilde{f}}_{i} ν$ if and only if $ν = {\tilde{e}}_{i} μ$ , with both sides defined simultaneously (Good-node deletion and cogood-node addition operators).

[F3]

The reachability proposition guarantees that a path of cogood additions from $\emptyset$ to every $p$ -restricted partition exists; the present example exhibits one explicitly and checks it node by node (Every $p$ -restricted partition is reached from the empty partition by cogood additions).

Verification

technique · direct

1.1givenF1

The $0$ -nodes of $\emptyset$ and $(1)$ . The empty partition has the single addable node $(1, 1)$ of residue $c - r = 0$ , so its residue- $0$ word is $(+)$ and the cogood node is $(1, 1)$ , whence ${\tilde{f}}_{0} \emptyset = (1)$ ; the partition $(1)$ has the single removable node $(1, 1)$ of residue $0$ , so its residue- $0$ word is $(-)$ with good node $(1, 1)$ and ${\tilde{e}}_{0} (1) = \emptyset$ .

1.2F1F2

The $1$ -nodes of $(1, 1)$ and $(1, 1, 1)$ . In $(1, 1)$ the residue- $1$ nodes are the addable nodes $(1, 2)$ (residue $2 - 1 = 1$ ) and $(3, 1)$ (residue $1 - 3 = - 2 \equiv 1$ ), ordered $(1, 2), (3, 1)$ top to bottom; the word is $(+, +)$ , the cogood node is the last surviving $+$ , namely $(3, 1)$ , and ${\tilde{f}}_{1} (1, 1) = (1, 1, 1)$ , while adding $(1, 2)$ would lead to $(2, 1)$ . In $(1, 1, 1)$ the residue- $1$ nodes are the addable node $(1, 2)$ and the removable node $(3, 1)$ ; the word $(+, -)$ is already reduced, so the surviving $+$ is $(1, 2)$ and the surviving $-$ is $(3, 1)$ : the cogood node is $(1, 2)$ with ${\tilde{f}}_{1} (1, 1, 1) = (2, 1, 1)$ , and the good node is $(3, 1)$ with ${\tilde{e}}_{1} (1, 1, 1) = (1, 1)$ , in agreement with the inverse property of [F2].

2.1step 1.1F1

The $2$ -nodes of $(1)$ and $(1, 1)$ . The residue- $2$ nodes of $(1)$ : the addable node $(2, 1)$ has residue $1 - 2 = - 1 \equiv 2$ , and there is no other, so the word is $(+)$ with cogood node $(2, 1)$ and ${\tilde{f}}_{2} (1) = (1, 1)$ . For $(1, 1)$ the residue- $2$ node is the removable node $(2, 1)$ (residue $1 - 2 \equiv 2$ ), the word is $(-)$ , the good node is $(2, 1)$ and ${\tilde{e}}_{2} (1, 1) = (1)$ .

2.2step 1.2F1F2

The $1$ -nodes of $(2, 1, 1)$ and the $2$ -nodes of $(2, 1, 1)$ and $(3, 1, 1)$ . In $(2, 1, 1)$ the residue- $1$ nodes are the removable nodes $(1, 2)$ (residue $2 - 1 = 1$ ) and $(3, 1)$ (residue $1 - 3 \equiv 1$ ), listed $(1, 2), (3, 1)$ : the word $(-, -)$ is reduced, so the good node is the first surviving $-$ , namely $(1, 2)$ , and ${\tilde{e}}_{1} (2, 1, 1) = (1, 1, 1)$ . The residue- $2$ nodes of $(2, 1, 1)$ are the single addable node $(1, 3)$ (residue $3 - 1 = 2$ ): the word is $(+)$ with cogood node $(1, 3)$ and ${\tilde{f}}_{2} (2, 1, 1) = (3, 1, 1)$ . In $(3, 1, 1)$ the only residue- $2$ boundary node is the removable node $(1, 3)$ ; the interior node $(2, 1)$ has the same residue but does not enter the signature; the word is $(-)$ with good node $(1, 3)$ and ${\tilde{e}}_{2} (3, 1, 1) = (2, 1, 1)$ .

3.1step 1.1step 2.1step 1.2step 2.2F2

The path and its reverse. Step 1.1, step 2.1, step 1.2 and step 2.2 exhibit, successively, $\emptyset \to (1) \to (1, 1) \to (1, 1, 1) \to (2, 1, 1) \to (3, 1, 1)$ as cogood additions with residues $0, 2, 1, 1, 2$ , and each reverse step as the deletion of the good node of the same residue; by [F2] this is exactly the assertion that each forward step is a cogood addition and each reverse step a good deletion, and the five residue labels match the displayed path.

4.1step 1.1step 2.1step 1.2step 2.2F3∎

All reduced signatures claimed in the example are the ones computed: $(-)$ for $(1)$ at $i = 0$ in step 1.1, $(-)$ for $(1, 1)$ at $i = 2$ in step 2.1, $(+, +)$ for $(1, 1)$ and $(+, -)$ for $(1, 1, 1)$ at $i = 1$ in step 1.2, and $(-, -)$ for $(2, 1, 1)$ at $i = 1$ , $(+)$ for $(2, 1, 1)$ at $i = 2$ and $(-)$ for $(3, 1, 1)$ at $i = 2$ in step 2.2; in particular $(-, -)$ exhibits two normal nodes of one residue and $(+, -)$ exhibits both signs in a reduced word, and the path still exists as [F3] predicts.

Depends on

Used by

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Sources

Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, Bull. Amer. Math. Soc. 47 (2010) 419-481, author arXiv version; §3.4 with the reduced signature (3.12)-(3.14), printed pp. 18-19 (the method of the computation). Complete 66-page text, SHA-256 8685199608967fa7c479e4dd883990b716a6f708e71cf75f6ce4aba41c6f2cab (standard reference, not scraped)