[Paper Review] On the Cartan matrix of Mackey algebras

Serge Bouc|arXiv (Cornell University)|Jan 2, 2010

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper provides explicit formulas for the determinant of the Cartan matrix of the Mackey algebra μk(G) and the rank of the Cartan matrix of the cohomological Mackey algebra coμk(G) over a field k of characteristic p > 0. It establishes that the Cartan matrix of coμk(G) is non-singular if and only if G is p-nilpotent with cyclic Sylow p-subgroups, and extends this to blocks: coμk(b) has non-singular Cartan matrix precisely when the block b is nilpotent with cyclic defect groups.

ABSTRACT

Let k be a field of characteristic p>0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra mu_k(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra comu_k(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of comu_k(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block comu_k(b) of comu_k(G) is non singular if and only if b is nilpotent with cyclic defect groups.

Motivation & Objective

• To derive an explicit formula for the determinant of the Cartan matrix of the Mackey algebra μk(G) over a field k of positive characteristic p.
• To determine the rank of the Cartan matrix of the cohomological Mackey algebra coμk(G), and characterize when it is non-singular.
• To generalize the characterization to blocks of coμk(G), identifying conditions under which the Cartan matrix of a block coμk(b) is non-singular.
• To establish a precise link between the algebraic structure of the Cartan matrix and group-theoretic properties such as nilpotency and cyclic defect groups.

Proposed method

• Utilizes the equivalence between the category of Mackey functors over R and the category of modules over the Mackey algebra μR(G), leveraging known structural properties of these algebras.
• Applies the decomposition of Mackey functors via p-perfect subgroups and the equivalence Mackk(G) ≅ ∏H Mackk(NG(H)/H, 1), focusing on the projective modules over μk(G, 1).
• Employs the Green ring ppk(G) of p-permutation kG-modules, with duality and Brauer quotient functors W ↦ W[Q] to analyze module structures and character relations.
• Uses induction and inflation functors to construct triangular transition matrices between character bases of p-permutation modules and cohomological Mackey functors.
• Applies the Brauer correspondence and block theory, particularly the structure of b-Brauer pairs (R, c), to relate the rank of the Cartan matrix to orbit counts on irreducible characters.
• Applies the inertial quotient and module extension techniques over algebraically closed fields to analyze the number of irreducible representations in blocks, especially in the nilpotent case.

Experimental results

Research questions

• RQ1 What is the explicit formula for the determinant of the Cartan matrix of the Mackey algebra μk(G) over a field of positive characteristic?
• RQ2 For which finite groups G is the Cartan matrix of the cohomological Mackey algebra coμk(G) non-singular?
• RQ3 How does the rank of the Cartan matrix of coμk(G) relate to the group-theoretic structure of G, particularly in terms of p-subgroups and p'-elements?
• RQ4 Under what conditions is the Cartan matrix of a block coμk(b) of the cohomological Mackey algebra non-singular?
• RQ5 What is the precise relationship between the nilpotency of a block b and the non-singularity of the Cartan matrix of coμk(b)?

Key findings

• The determinant of the Cartan matrix of μk(G) is given by an explicit formula involving the number of conjugacy classes of p'-elements and cyclic p-subgroups.
• The rank of the Cartan matrix of coμk(G) is equal to the number of conjugacy classes of pairs (R, s), where R is a cyclic p-subgroup of G and s is a p'-element in the centralizer of R.
• The Cartan matrix of coμk(G) is non-singular if and only if G is p-nilpotent with cyclic Sylow p-subgroups.
• For a block b of kG, the Cartan matrix of coμk(b) is non-singular if and only if b is nilpotent with cyclic defect groups.
• When b is nilpotent with cyclic defect group D, the number of irreducible characters of kNG(R)BrR(b) equals the number of NG(R)-orbits on Irr(kCG(R)BrR(b)) for each cyclic p-subgroup R.
• In the nilpotent case with cyclic defect, the inertial quotient is trivial (order 1), which ensures the equality of orbit counts and thus non-singularity of the Cartan matrix.