DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cell data, multipartition dominance, and higher-cell ideals

Definition

Let $R$ be a commutative ring and let $A$ be a unital associative $R$ -algebra. A cell datum for $A$ consists of a finite poset $(Λ, \geq)$ , finite sets $T (λ)$ for $λ \in Λ$ , an $R$ -basis ${C_{s, t}^{λ} : λ \in Λ, s, t \in T (λ)}$ of $A$ , and an $R$ -linear involutive algebra anti-automorphism $*$ satisfying $(C_{s, t}^{λ})^{*} = C_{t, s}^{λ}$ and the following coefficient-independence axiom: for every $λ \in Λ$ , $s, t \in T (λ)$ , and $a \in A$ , there are coefficients $r_{t, v}^{λ} (a) \in R$ , independent of $s$ , such that $C_{s, t}^{λ} a \equiv \sum_{v \in T (λ)} r_{t, v}^{λ} (a) C_{s, v}^{λ} (m o d A^{> λ}),$ where $A^{> λ}$ is the $R$ -span of basis elements $C_{u, v}^{μ}$ with $μ > λ$ . Write $A^{\geq λ}$ for the span with $μ \geq λ$ .

For multipartitions $λ, μ$ of $n$ with $r$ ordered components, write $λ ⊵ μ$ if for every component $1 \leq c \leq r$ and row cutoff $j \geq 0$ , $\sum_{d < c} ∣ λ^{(d)} ∣ + \sum_{i = 1}^{j} λ_{i}^{(c)} \geq \sum_{d < c} ∣ μ^{(d)} ∣ + \sum_{i = 1}^{j} μ_{i}^{(c)},$ with missing rows padded by zero. Write $λ ▹ μ$ when $λ ⊵ μ$ and $λ \neq μ$ . The row-standard initial multitableau $t^{λ}$ fills component $1$ row by row, then component $2$ row by row, and so on, using $1, \dots, n$ in order.

Facts & Assumptions

Given: A commutative ring $R$ , a unital associative $R$ -algebra $A$ , and the finite multipartition data described in the Definition.

[F1]

An $R$ -algebra has a unital ring structure with a central structural image of $R$ . (Algebras over a commutative ring, central structure maps, and algebra homomorphisms)

[F2]

An $r$ -multipartition is an $r$ -tuple of partitions of total size $n$ , and its standard multitableaux are the linear extensions of the componentwise box poset. (Multipartitions, standard multitableaux, and generic contents)

Verification

technique · direct

1.1givenF2algebra

The dominance inequalities are reflexive and transitive. If they hold in both directions, all displayed cumulative sums are equal; subtracting successive row cutoffs and then successive component prefixes recovers every row length in every component, so $λ = μ$ . Thus $⊵$ is a partial order. The tableau $t^{λ}$ is standard: its entries increase left to right by construction, and every box in an earlier row or earlier component is filled before every box in a later row or component, so each column also increases top to bottom.

2.1givenF1step 1.1algebra

Fix $λ$ . If $μ > λ$ , the cell axiom applied to $C_{u, v}^{μ} a$ puts it in the span of the shape- $μ$ basis elements plus $A^{> μ}$ . Every shape in that expression is strictly above $λ$ by transitivity; hence $A^{> λ}$ is a right ideal. Since $*$ preserves each shape and $A^{> λ}$ , $a x = (x^{*} a^{*})^{*}$ lies in $A^{> λ}$ whenever $x$ does, so it is also a left ideal. The same argument with $μ \geq λ$ proves that $A^{\geq λ}$ is a two-sided ideal.

3.1givenstep 2.1∎

The given basis of $A^{\geq λ}$ is the disjoint union of the shape- $λ$ basis and the basis of $A^{> λ}$ . Linear independence therefore makes the classes of $C_{s, t}^{λ}$ a basis of the quotient $A^{\geq λ} / A^{> λ}$ . No choice is needed because the poset and every indexing set are finite.

Depends on

Used by

Cyclotomic Murphy elements Definition
Rank-one cyclotomic cell modules and their forms Example
Rank-two type-A cell modules at a singular parameter Example
Cyclotomic component shift and multipartition dominance ideals Lemma
Integral type-A Garnir straightening and the Murphy cell law Lemma
Cell modules, their forms, and radicals Theorem
Nonzero cellular heads classify the simple modules over a field Theorem
The cyclotomic Murphy basis and cellularity Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete author-uploaded preliminary lecture manuscript (1998) (standard reference, not scraped)
Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic q-Schur algebras, Mathematische Zeitschrift 229 (1998), 385–416 (standard reference, not scraped)