[Paper Review] A cellular algebra with certain idempotent decomposition

Kentaro Wada|ArXiv.org|May 8, 2008

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper introduces a generalized framework for cellular algebras by decomposing the unit element into orthogonal idempotents compatible with a cellular basis, enabling the construction of Levi-type and parabolic-type subalgebras. It establishes isomorphisms between decomposition numbers of the original algebra and a quotient algebra, and classifies blocks via a refined block decomposition, extending prior results on cyclotomic q-Schur algebras to general cellular algebras.

ABSTRACT

For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\a$, we give a coarse partition of $\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\aA$ of $\A$ and its quotient algebra $\oA$, and also construct a parabolic type subalgebra $ A$ of $\A$, which contains $\aA$ with respect to the map $\a$. Then, we study the relation of standard modules, simple modules and decomposition numbers among these algebras. Finally, we study the relationship of blocks among these algebras.

Motivation & Objective

• To generalize the product formula for decomposition numbers in cyclotomic q-Schur algebras to arbitrary cellular algebras with compatible idempotent decompositions.
• To construct a Levi-type subalgebra and a parabolic-type subalgebra from a given cellular algebra using a map α.
• To relate standard modules, simple modules, and decomposition numbers across the original algebra and its subquotients.
• To classify blocks of the original algebra and its subalgebras using the induced block decomposition from the map α.
• To extend the representation-theoretic framework of cellular algebras by incorporating non-cellular parabolic-type subalgebras.

Proposed method

• Decompose the unit element of a cellular algebra into orthogonal idempotents satisfying compatibility conditions with the cellular basis.
• Use a map α to induce a coarse partition of the cellular basis, refining the original partition from the idempotent decomposition.
• Construct a Levi-type subalgebra 𝔸^α and a parabolic-type subalgebra ~𝔸^α containing 𝔸^α, both derived from the cellular basis and the map α.
• Define a quotient algebra 𝔸̅^α as a quotient of ~𝔸^α, which inherits a simpler structure analogous to the quotient in cyclotomic q-Schur algebras.
• Establish isomorphisms between decomposition numbers of 𝔸 and 𝔸̅^α, showing that certain decomposition numbers of the original algebra coincide with those of the quotient.
• Classify blocks of 𝔸, ~𝔸^α, 𝔸^α, and 𝔸̅^α using the induced block decomposition via the map α and the equivalence relation ∼ on the poset Λ⁺.

Experimental results

Research questions

• RQ1 How can the decomposition numbers of a general cellular algebra be related to those of a simpler quotient algebra constructed via idempotent decomposition and a map α?
• RQ2 What is the structural relationship between the original cellular algebra, its Levi-type subalgebra, parabolic-type subalgebra, and the quotient algebra?
• RQ3 Can the block structure of a cellular algebra be recovered from the block structures of its subalgebras and quotient algebra via the map α?
• RQ4 Does the quotient algebra 𝔸̅^α inherit a cellular structure that allows for a block decomposition in terms of the original algebra’s blocks?
• RQ5 Under what conditions does the map α induce a compatible decomposition of standard and simple modules across the algebraic hierarchy?

Key findings

• The decomposition number of a standard module W^λ in the original algebra 𝔸 coincides with the corresponding decomposition number in the quotient algebra 𝔸̅^α for certain λ and μ.
• The quotient algebra 𝔸̅^α is isomorphic to a direct sum of tensor products of smaller cellular algebras, generalizing the structure observed in cyclotomic q-Schur algebras.
• The block decomposition of 𝔸̅^α is given by ⨁_{Γ∈Λ⁺/~} ⨁_{η∈X⁺} 𝔹̅^α_{Γ_η}, where Γ_η is the set of λ ∈ Γ with α(λ) = η.
• The block 𝔹̅^α_Γ of 𝔸̅^α decomposes as a direct sum over η ∈ X⁺, indicating a refined block structure induced by the map α.
• The parabolic-type subalgebra ~𝔸^α is not cellular but is standardly based, and its block structure is linked to that of 𝔸 via the map α.
• The Levi-type subalgebra 𝔸^α and the quotient algebra 𝔸̅^α are both cellular, and their module categories are related to those of the original algebra through the induced map α.