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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 suprising 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?

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