What is the correct formulation of the CDE triangle?

Question

The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as people must have considered this.

For finite groups we are interested in comparing representations in characteristic $p$ and in characteristic zero. A finite group algebra in characteristic zero is semi-simple. Therefore one can generalise this to finite dimensional algebras over a local ring (which are free as a module) and such that the algebra over the field of fractions is semisimple.

I would like to go further and drop the condition that the algebra over the field of fractions is semisimple. Then the triangle should be replaced by a square. In the semisimple case one edge of this square is the identity matrix (the Cartan matrix of a semisimple algebra).

Once this has been done I would like a statement that says this is compatible with composing specialisations.

As a motivating example take the Hecke algebras over the ring of Laurent polynomials, $\mathbb{Z}[q,q^{-1}]$. Then we have homomorphisms to the fields $\mathbb{Q}(q)$, $\mathbb{Q}(\omega)$ (taking $q$ to a root of unity) and to a finite field. This gives two decomposition matrices in the usual sense.

Is there also a decomposition matrix going from $\mathbb{Z}[q,q^{-1}]$ to $\mathbb{Z}(\omega)$? and does this give a factorisation of the decomposition matrix going from $\mathbb{Z}[q,q^{-1}]$ to the finite field?

Answer 1

I would say that the correct general formulation of the CDE is as follows: let $A$ be a complete local ring with fraction field $K$ and residue field $k$. Let $R$ be an $A$-algebra which is free and finite rank. Let $R_k$ and $R_K$ be the base changes. Then we have the Cartan matrix $C$ of $R_k$, whose entries are multiplicities of simple modules in projective covers.

We also have the decomposition matrix, which gives the multiplicities of simple $R_k$-modules in the base change of the A-form of a simple $R_K$-modules (note that this A-form is a choice, but hopefully the resulting multiplicities don't depend on it).

Finally, we can take a projective $R_k$ module; since this has trivial self-exts, we can deform this to an R-module in a unique way. I think I'm really using completeness here; my intuition is that local isn't good enough, but maybe one of the experts out there can clarify. The usual $E$ matrix would be the multiplicities of the simples of $R_K$ in the base change of this deformation. I believe you're right that this deformation must be projective (given any $R$-module, you can think of the Ext's as a coherent sheaf on Spec A, and its stalk vanishes at the only closed point), and so one could also incorporate the Cartan matrix of $R_K$ into this picture.

The result is a factorization $C_k=DC_KE$ where now $E$ is defined in terms of multiplicities of indecomposible projectives in the deformation.

In the example you wrote, I haven't thought extremely carefully, but I'm worried that you haven't taken completions.

Answer 2

To revisit Bruce's earlier question, it might be useful to suggest a more sceptical alternative to Ben's answer and the related comments. I doubt that there will be a "correct" version of the CDE-triangle that has enough breadth to take in the variety of analogues that have emerged in representation theory, though it's obviously important to make the assumptions precise about underlying fields, local rings, etc. See also the conference paper: MR2184010 (2006g:17027) 17B67 (17B10) Ekedahl, Torsten (S-STOC), Kac-Moody algebras and the cde-triangle. Noncommutative geometry and representation theory in mathematical physics, 49–58, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005.

For me the essential ingredients to start with are suitable module categories (or more general analogues) in which three distinctive types of objects play a leading role: (1) simple modules, which are "small" and basic but often hard to get at directly; (2) indecomposable projectives (or injectives ...) which are also naturally present in suitable categories and usually have just finitely many composition factors, but are typically "large" and messy to study; (3) intermediate objects, easier to construct and often having known dimensions (if finite) or "formal" characters. The idea then is to express (say) a projective cover of a simple module formally, perhaps via a filtration, in terms of the intermediate objects. To make this interesting, when all composition factor multiplicities are finite or otherwise controllable, one wants a kind of "reciprocity" between multiplicities of simples in intermediate objects and multiplicities of the latter in the big modules. Here is some brief history followed by a sample of more recent instances. (It does get a bit long...)

(1) Elie Cartan actually studied what we now call finite dimensional associative algebras and emphasized the importance of knowing the composition factor multiplicities of indecomposable projectives (now dubbed Cartan invariants). For a finite group with $r$ classes of elements having orders not divisible by a given prime $p$, you get an $r \times r$ matrix $C$ (over a large enough field).

(2) Work by Richard Brauer and his Toronto student Cecil Nesbitt after 1937 introduced in the finite group setting (for a prime $p$ dividing the group order and large enough fields) an $s \times r$ decomposition matrix $D$ showing how to express the $s$ ordinary irreducible characters (= number of classes) as formal sums of $p$-modular irreducible characters (counting composition factor multiplicities in reduction mod $p$ for any suitable lattice in a module). Then $C = D^{t} D$ (so $C$ is symmetric), where the transpose of $D$ shows how to express a projective (lifted to characteristic 0) as a combination of ordinary characters. These ideas were exposed by Curtis-Reiner (1962) in Section 83, etc. Brauer was studying $p$-blocks, which show up in the block decompositions of the matrices.

(3) Following Swan's formalism, Serre (1971) formulated the more abstract cde-triangle in his part III, using homomorphisms between various Grothendieck groups. This was further codified by Curtis-Reiner in their later 1981 book, Section 18, with a lot of attention to the rings and fields involved.

(4) Having studied the old CR book in a 1963-64 course at Yale taught by Jacobson (!), I later tried to adapt $C = D^{t} D$ to modular representations of Lie algebras of simple algebraic groups (working just in prime characteristic). The f.d. restricted enveloping algebra imitates a finite group algebra in some ways. Here the "intermediate" objects were f.d. analogues of Verma modules. At first I studied blocks and multiplicities but not filtrations. This rough version appeared in J. Algebra (1971) and inspired Verma's introduction of affine Weyl groups relative to $p$, as well as much more sophisticated work by Jantzen treating filtrations of projectives, plus action of a maximal torus in the group.

(5) Work by Bernstein-Gelfand-Gelfand in the early 1970s was partly inspired by Jantzen's work and by my 1971 paper, leading to their "BGG category" and "BGG reciprocity" in 1976. Then Kazhdan-Lusztig theory for finite and affine Weyl groups came into play, etc.

(6) Eventually some but not all of the ideas spread elsewhere in representation theory, including the work by Alvany Rocha and her thesis advisor Nolan Wallach and much other work on Kac-Moody algebras (recently by Arakawa-Fiebig for the mysterious critical level in the affine case). Plus early work by Dan Nakano on other modular Lie algebra settings, recent work on rational Cherednik algebras, and so on.

(7) The most fruitful general formulation for some purposes was given in a 1988 Crelle paper by Cline-Parshall-Scott on highest weight categories and quasi-hereditary algebras. Other general settings were proposed by Ron Irving and by Apoorva Khare. It's hard though to find just one common framework.