Residue signatures, normal, good, conormal and cogood nodes

Let $p$ be a prime and let $\mu$ be a $p$-restricted partition, so that ${\mu }_i-{\mu }_{i+1}<p$ for all $i\ge 1$ (p-regular and p-restricted partitions, Partitions, English diagrams, and conjugation). A node $(r,c)$ of the Young diagram $[\mu ]$ is an $i$-node if its residue $\mathrm{res}(r,c):=c-r<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p$ equals $i\in \mathbb{Z}\mathrm{/}p$; the $i$-nodes of $\mu$ are its addable and removable nodes of residue $i$ (Removable and addable nodes).

The reduced $i$-signature. Fix $i\in \mathbb{Z}\mathrm{/}p$ and list the $i$-nodes of $\mu$ as $A_1,\dots ,A_n$ from top to bottom, that is, in increasing row index and, for equal row index, in increasing column index. Form the word $\sigma =({\sigma }_1,\dots ,{\sigma }_n)$ with ${\sigma }_m=+$ if $A_m$ is addable and ${\sigma }_m=-$ if $A_m$ is removable. Whenever $r<s$, ${\sigma }_r=-$, ${\sigma }_s=+$ and ${\sigma }_{r+1}=\cdots ={\sigma }_{s-1}=0$, replace ${\sigma }_r$ and ${\sigma }_s$ by $0$; repeat until no $-$ lies to the left of a $+$. The resulting reduced $i$-signature has the form $+\cdots +-\cdots -$ with the zeros deleted, and the nodes attached to its signs are:

• the normal $i$-nodes of $\mu$ are the removable nodes corresponding to the $-$ signs of the reduced signature;
• the good $i$-node is the topmost (first listed) of the normal $i$-nodes;
• the conormal $i$-nodes of $\mu$ are the addable nodes corresponding to the $+$ signs of the reduced signature;
• the cogood $i$-node is the bottommost (last listed) of the conormal $i$-nodes.

Write ${\epsilon }_i(\mu )$ and ${\phi }_i(\mu )$ for the numbers of normal and of conormal $i$-nodes of $\mu$; the good $i$-node exists exactly when ${\epsilon }_i(\mu )>0$, and the cogood $i$-node exactly when ${\phi }_i(\mu )>0$. At most one good and at most one cogood $i$-node occur, for each residue.

Well-definedness. Each reduction step deletes two nonzero entries, so any sequence of steps terminates after at most $\lfloor n\mathrm{/}2\rfloor$ steps, and in the resulting word no $-$ lies to the left of a $+$: otherwise, take two nonzero entries that are consecutive among the nonzero entries of the current word and are ordered $-$, $+$; all entries strictly between them are $0$ by the choice, so the pair would still be reducible. Hence, after deleting the zero entries, a terminal word is ${+}^a{-}^b$ for some $a,b\ge 0$; cancelled positions remain in their original places and need not form a contiguous block. The terminal word does not depend on the order in which reducible pairs are chosen: two reducible pairs cannot overlap partially, because an endpoint of one pair lying strictly inside the other would be a nonzero entry in a position that the other pair requires to be $0$, and pairs sharing exactly one endpoint would similarly place a nonzero entry strictly inside the other pair; so any two reducible pairs are either disjoint (and the corresponding steps commute) or equal, there are no critical pairs, and the rewriting relation is locally confluent. Uniqueness follows by induction on the number of nonzero entries. A terminal word admits no step. Otherwise, two distinct first steps commute and reach a common word after one further step each. All these successor words have fewer nonzero entries, so the induction hypothesis identifies their terminal words. Thus any two reduction sequences have the same terminal word; in it the first surviving $-$ and the last surviving $+$ are unique positions whenever they exist. Thus the reduced signature, the sets of normal and conormal nodes, and the good and cogood nodes are well defined. The empty partition has the unique addable node $(1,1)$, of residue $0$, and no removable nodes. Its $0$-signature is $(+)$ and its other signatures are empty. Thus ${\epsilon }_i(\mathrm{\varnothing })=0$ for every $i$, ${\phi }_0(\mathrm{\varnothing })=1$, and ${\phi }_i(\mathrm{\varnothing })=0$ for $i\mathrm{\ne }0$. It has no good node and its cogood $0$-node is $(1,1)$.

Convention warning. Brundan's $\mathrm{\ell }$-residue $r-c$ is the negative of our $c-r$. His normal and good removable nodes (§§2.2–2.4) are the same nodes on the same partition as ours, with residue names negated. Indeed, a minus survives our cancellation exactly when each prefix of the boundary word strictly below it has at least as many minuses as pluses. Brundan's lattice test expresses these inequalities after the noncorner entries common to his demand and support sets are removed, as in Remark 2.4(I). Remark 2.4(II) compares his convention with Kleshchev's older convention on the transpose; it does not identify our nodes with Brundan-normal nodes on the transpose. Transposition and residue negation must therefore be tracked separately (Removable and addable nodes, Partitions, English diagrams, and conjugation).

• Zero entries are not nodes. The zeros of the word record cancelled places and carry no node; only the surviving signs are attached to nodes, as in the definition of the normal, good, conormal and cogood nodes.
• Why $p$-restrictedness is assumed. The nodes and their residues are defined for every partition, but the definition is applied on this page to the $p$-restricted labels, for which the good-node machinery matches the modular branching rules of the dual-Specht convention used throughout.
• At most one sign of each outer type. In the zero-suppressed reduced word ${+}^a{-}^b$ the first surviving $-$ (if $b>0$) and the last surviving $+$ (if $a>0$) are unique places, which is the content of the final assertion.
• Choice. The definition and the well-definedness argument use only finite lists and a finite reduction; no choice principle is used.