Cyclotomic Murphy Bases and Cellular Module Constructions

Cellular structure organizes representations at singular parameters, where generic eigenvector formulas no longer apply. The Murphy basis is an integral basis theorem with a triangular multiplication law and involution, and the cell-module quotient constructions must be justified independently of generic semisimplicity.

This is a prose scaffold for future item authoring. The constructions and results below are explicit proof obligations; an empty item list does not certify that they have been proved in the library. The source-grounded contracts and prerequisite audit are recorded in research/plan-hopf-hecke-algebras-track.md.

Construction and proof obligations

def-hh-cell-datum-dominance-and-cell-ideals. Define a finite poset, basis C_st^λ, anti-involution *, and the coefficient-independence axiom modulo the higher-cell span. Prove the higher-cell span is a two-sided ideal from this axiom before forming quotients. Give multipartition dominance and row-standard tλ explicitly.

def-hh-cyclotomic-murphy-elements. For a_c=Σ_(b<c)|λ^(b)| set u_λ=∏(c=2)^r∏(i=1)^a_c(L_i−u_c) and x_λ=Σ_(w∈S_λ)T_w, with S_λ the rows of the initial standard multitableau. Prove x_λu_λ=u_λx_λ. Set m_λ=u_λx_λ and m_st=T_(d(s))^*m_λT_(d(t)), where d(t) sends the initial tableau to t and * fixes T_i,L_i. These are genuine elements before any basis/cellularity claim; no omitted parameter rescaling is allowed.

lem-hh-type-a-garnir-straightening-and-murphy-cell-law. Supply the full ordinary type-A integral proof locally (Mathas §§3.3–3.18, printedpp18–25), using the earlier noncircular HH-12 basis. For adjacent-row Garnir belt (i,j), split λ into ν=(λ_1,…,λ_(i−1),j−1,λ_i+1,λ_(i+1)−j,λ_(i+2),…) and equate the two length-additive coset expansions of Σ_(x∈S_ν S_λ)T_x: Σ_(w∈S_ν∩D_λ)x_λ T_w=Σ_(v∈S_λ∩D_ν)T_v^*x_ν. Isolate the unique Garnir term to express x_λT_d(g) as larger-shape row terms minus strictly more dominant standard right-tableau terms. Every nonstandard row tableau factors g w with additive length by reducing its inversion number while retaining a column violation. Straighten in the finite lexicographic shape/tableau order, on each side. The equal-cardinality square spanning matrix and HH-12 rank n! prove basis. Establish the column eigenvalue −1 refinement and use row eigenvalues Q to eliminate coefficients of shapes failing dominance (Mathas Prop3.16), proving dominance ideals and the fixed-initial-row right multiplication law (Cor3.17/Thm3.18). Lexicographic shape growth alone is not called dominance. For the dominance upgrade, propagate a nonzero coefficient across valid adjacent swaps within each row-label block using the universal row-eigenvalue equations. Linear extensions of a finite poset are connected by such swaps; every cover pair can be adjacent in some extension. If two block nodes share a column, a neighboring vertical cover pair can therefore be assigned consecutive block labels, contradicting the proved column eigenvalue −1 versus Q. Each block consequently forms a horizontal strip; the first p blocks have at most p boxes per column, forcing Σ_(j≤p)μ_j≥Σ_(j≤p)λ_j. Carry this universal-domain argument through basis specialization rather than cancel Q+1 at singular parameters.

lem-hh-componentwise-initial-row-straightening. Use the fixed-initial-row type-A cell law in each component interval and the commuting factor u_λ; no stronger simultaneous tableau inequality is assumed from an unread source. Decompose row permutations into within-component permutations followed by distinguished component coset representatives, with additive length. These coset representatives preserve the order inside each component. Tableau-prefix dominance is preserved under their relabeling: for each cutoff m, the selected labels in each component form an initial segment, so every cumulative-row inequality becomes the already proved inequality at that segment size. This elementary counting replaces DJM3.17’s imported Bruhat cancellation. Componentwise shape dominance lifts to multipartition dominance; modulo higher shapes, multiplying m_(tλ,t) by finite T_i retains the same initial left row. Left multiplication and * give the shape-ideal closure needed by the next lemma.

lem-hh-cyclotomic-component-shift-and-dominance-ideals. For a=(a_1,…,a_r), u_a=∏c∏(j≤a_c)(L_j−u_c), b=a+e_k, prove u_a T_(a_k)⋯T_1 X_1 T_1⋯T_(a_k)=Q^(a_k)u_k u_a+Q^(a_k)u_b by L_(a_k+1)=Q^(−a_k)T_(a_k)⋯T_1 X_1 T_1⋯T_(a_k). Right-multiply by the proved inverse T products. For m_(tλ,t)X_1 decompose d(t) into a component permutation and component coset, then separate the term with u_a from that with u_b. The first is governed by the previous fixed-row law. The second moves one first-row node of component k to a new final row of component k−1; express x_λ via its length-additive row coset sum x_νΣ T_c. This raises multipartition dominance; at k=1 the product u_b contains ∏(L_1−u_c) and is zero. Apply ordinary component straightening to restore partitions. Induction over generator words proves every higher-shape span is a right ideal; * proves two-sidedness. Modulo higher shapes, initial-left-row closure holds for X_1 as well. This is the complete DJM3.4/3.15–3.25 route; no cyclotomic PBW or type-A Murphy theorem is left as an external supplier.

thm-hh-cyclotomic-murphy-basis-and-cellularity. Use the three earlier integral suppliers: ordinary type-A Garnir/Murphy straightening, componentwise initial-row straightening, and the u_a/T_0 dominance-raising identity. Together they prove the spans N^≥λ,N^>λ are two-sided ideals and (m_(tλ,t))H reduces modulo N^>λ to the same fixed initial row, with coefficients independent of any later left multiplication. Thus m_st h≡Σ_v r_(t,v)(h)m_sv mod N^>λ and * swaps indices. The shape (empty,…,1^n) has m_λ=1, so this fixed-row closure proves all standard m_st span H. HH-16 colored RSK counts exactly r^n n! standard pairs, equal to the proved integral PBW rank. The coordinate matrix of this spanning family is square and has a right inverse; its determinant is a unit, so the family is an integral basis. This completes cellularity over universal R, and tensoring gives every specialization. Generic diagonalization is not a substitute for this integral argument.

thm-hh-cell-modules-forms-and-radicals. Construct the free cell module using the stated coefficient matrices, prove module laws by quotient associativity, define its bilinear form from multiplication, and prove independence, symmetry and invariance. Radical is a submodule by invariance; quotient is well-defined.

thm-hh-cellular-simple-module-classification-over-a-field. Prove a nonzero cell-form quotient is absolutely simple by the rank-one action identity; prove all simple modules arise by choosing a minimal nonannihilating cell ideal, and prove distinct nonzero quotients are nonisomorphic. All steps use cell ideals/finite poset induction locally, not Ariki or KLR categorification.

Reading and applications

Prerequisite pages: tensor-coherence-and-algebraic-descent, coxeter-presentations-exchange-and-reduced-word-theorems, generic-coxeter-hecke-algebras-and-the-standard-basis, type-a-affine-hecke-algebras-and-bernstein-pbw, cyclotomic-hecke-quotients-and-ariki-koike-pbw, chain-conditions-and-semisimple-modules. The companion cyclotomic-murphy-bases-and-cellular-module-constructions-examples develops the calculations and failures needed to test these constructions.