The Cartan Matrix | KEVOS® Mathematics

Q: What does C = 1 tell me?

That the Cartan map K_0R → G_0R is an isomorphism, so every finitely generated module has a well-defined class expressible in terms of projectives. This holds for rings of finite global dimension, such as T_n(k) and the first-row algebra, and fails for Z/p^mZ with m ≥ 2.

Executive Summary

Let $R$ be right artinian, $e_{1} R, \dots, e_{r} R$ a complete list of principal indecomposables and $V_{i} = e_{i} R / e_{i} J$ the corresponding simple right modules. Every $e_{i} R$ has a composition series, and the Cartan invariant $c_{i j}$ counts the factors isomorphic to $V_{j}$ . The matrix $C = (c_{i j}) \in M_{r} (ℤ)$ is the Cartan matrix of $R$ .

It is a compact record of how far $R$ is from semisimple: $C$ is the identity exactly when all principal indecomposables are simple. Its diagonal entries are at least $1$ , its $i$ -th row sums to the length of $e_{i} R$ , it is block diagonal along the blocks of $R$ , and it is the matrix of the natural map $K_{0} R \to G_{0} R$ .

$c_{i j}$ Factors of

e_{i} R

of type

V_{j}

$c_{i i} \geq 1$ Diagonal entries

row sumLength of

e_{i} R

$K_{0} \to G_{0}$ The Cartan map

Overview

Two finite lists attach to a right artinian ring: the projective indecomposables and the simple modules. They have the same length $r$ , and $e R \mapsto e R / e J$ matches them up. The Cartan matrix measures the failure of that match to be an equality of modules — it records what else is inside $e_{i} R$ besides its top.

e_{i} R ⊋ e_{i} J \supseteq \dots \supseteq 0, c_{i j} = [e_{i} R : V_{j}],

(25.C)

The bracket denotes the multiplicity of $V_{j}$ among the composition factors, well defined by the Jordan–Hölder theorem.

Right artinian-ness is what makes the definition legitimate: it guarantees that finitely generated modules have composition series. Semiperfectness alone is not enough, and there are semiperfect rings where the right Cartan matrix exists and the left one does not.

Right modules, following §25. The transpose convention and the left-right convention both vary across the literature; see the notation section before comparing a Cartan matrix with a published one.

Learning Objectives

State the hypotheses under which the Cartan matrix is defined.
Read off lengths, diagonal lower bounds and block structure from $C$ .
Prove the criterion $c_{i j} \neq 0 \Leftrightarrow e_{i} R e_{j} \neq 0$ from the composition factor test $(21.19)$ .
Compute $C$ for $T_{n} (k)$ and for the first-row algebra, and compare with the left Cartan matrix.
Interpret $C$ as the matrix of the Cartan map $K_{0} R \to G_{0} R$ and $\det C$ as the index of its image.
Quote the Brauer–Nesbitt–Nakayama theorem for group algebras with its hypotheses.

Definitions

Definition§25Cartan matrix

Let $R$ be a right artinian ring with $J = rad R$ . Choose a complete irredundant list $e_{1} R, \dots, e_{r} R$ of principal indecomposable right $R$ -modules and set $V_{i} = e_{i} R / e_{i} J$ , so that $V_{1}, \dots, V_{r}$ is a complete irredundant list of simple right $R$ -modules. Define $c_{i j} \geq 0$ to be the number of composition factors of $e_{i} R$ isomorphic to $V_{j}$ . The matrix $C = (c_{i j}) \in M_{r} (ℤ)$ is the right Cartan matrix of $R$ .

Well-definedness: $e_{i} R$ is finitely generated, hence of finite length over a right artinian ring; Jordan–Hölder makes the multiplicities independent of the chosen composition series.
Ambiguity: Reordering the principal indecomposables replaces $C$ by $P C P^{- 1}$ for a permutation matrix $P$ . The Cartan matrix is an invariant only up to that conjugation.
Left Cartan matrix: The same construction with $R e_{i}$ and $R e_{i} / J R e_{i}$ ; it requires $R$ to be left artinian and is in general a different invariant.
$K_{0} R$ , $G_{0} R$: The Grothendieck groups of finitely generated projective modules and of all finitely generated modules; both are free abelian of rank $r$ for $R$ right artinian.

Core Concepts

Testing for a composition factor

The computational key is a criterion from the idempotent theory: for a local idempotent $e$ and a right $R$ -module $M$ of finite length, $V = e R / e J$ occurs as a composition factor of $M$ if and only if $M e \neq 0$ , if and only if ${Hom}_{R} (e R, M) \neq 0$ . Applied to $M = e_{i} R$ and $e = e_{j}$ this converts a statement about composition series into a statement about a corner of the ring.

c_{i j} \neq 0

V_{j}

a factor of

e_{i} R

(e_{i} R) e_{j} \neq 0

e_{i} R e_{j} \neq 0

Why $C$ splits along blocks

If $e_{i}$ and $e_{j}$ lie in different blocks then $e_{i} R e_{j} = 0$ , so $c_{i j} = 0$ . Ordering the idempotents block by block therefore puts $C$ into the shape $diag (C_{1}, \dots, C_{s})$ , one diagonal block per block of $R$ , and $C_{m}$ is precisely the Cartan matrix of the block $B_{m}$ regarded as a ring in its own right.

Each $C_{m}$ is indecomposable as a matrix: no relabelling of the idempotents within a block splits it further, because within a block any two indices are joined by a chain of indices with consecutive nonzero entries — that is exactly what linkage says. So $diag (C_{1}, \dots, C_{s})$ is the finest block diagonalisation available.

The K-theoretic reading

For $R$ right artinian, $K_{0} R$ is free abelian with basis the classes $[e_{i} R]$ , by uniqueness of decomposition of finitely generated projectives, and $G_{0} R$ is free abelian with basis the classes $[V_{j}]$ , by Jordan–Hölder. The map $c : K_{0} R \to G_{0} R$ that forgets projectivity sends $[e_{i} R]$ to $\sum_{j} c_{i j} [V_{j}]$ , so $C$ is its matrix.

Key Results

Proposition(21.19)Composition factor criterion

Let $R$ be a ring, $e \in R$ a local idempotent, $J = rad R$ , and let $M$ be a right $R$ -module of finite composition length. Then $M$ has a composition factor isomorphic to $e R / e J$ if and only if $M e \neq 0$ , if and only if ${Hom}_{R} (e R, M) \neq 0$ .

Corollary—Support of the Cartan matrix

Let $R$ be right artinian with primitive idempotents $e_{1}, \dots, e_{r}$ representing the principal indecomposables. Then $c_{i j} \neq 0$ if and only if $e_{i} R e_{j} \neq 0$ . Consequently $c_{i j} = c_{j i} = 0$ whenever $e_{i}$ and $e_{j}$ lie in different blocks of $R$ .

Proof

Apply the previous proposition to $M = e_{i} R$ , which has finite length because $R$ is right artinian, and to the local idempotent $e_{j}$ : $V_{j}$ is a composition factor of $e_{i} R$ exactly when $(e_{i} R) e_{j} = e_{i} R e_{j} \neq 0$ . For the second assertion, if $e_{i}$ and $e_{j}$ lie in blocks $c_{m} R$ and $c_{m'} R$ with $m \neq m'$ , then $e_{i} R e_{j} = e_{i} c_{m} R c_{m'} e_{j} = 0$ because $c_{m} R c_{m'} = 0$ for distinct centrally primitive idempotents.

Theorem§25Block diagonal form

Let $R$ be right artinian with block decomposition $R = B_{1} \oplus \dots \oplus B_{s}$ . Index the principal indecomposables so that those belonging to a common block are consecutive. Then $C = diag (C_{1}, \dots, C_{s})$ , where $C_{m}$ is the Cartan matrix of the ring $B_{m}$ ; each $C_{m}$ is indecomposable, so this is the finest block diagonal decomposition of $C$ obtainable by relabelling.

Proof

Vanishing off the diagonal blocks is the previous corollary. Within a block, the principal indecomposables of $R$ lying in $B_{m}$ are exactly the principal indecomposables of the ring $B_{m}$ , and their composition factors as $R$ -modules coincide with their composition factors as $B_{m}$ -modules because the other blocks act as zero; hence the diagonal block is the Cartan matrix of $B_{m}$ . Indecomposability of $C_{m}$ follows from the definition of linkage: any two indices in a block are connected by a chain $i = i_{0}, i_{1}, \dots, i_{l} = j$ with $e_{i_{q}} R e_{i_{q + 1}} \neq 0$ or $e_{i_{q + 1}} R e_{i_{q}} \neq 0$ , so no partition of the index set into two nonempty parts makes all crossing entries vanish.

Proposition§25, Ex. 1–2Cartan invariants as corner dimensions

Let $A$ be a finite-dimensional algebra over a field $k$ which is a splitting field for $A$ , that is, ${End}_{A} (V) = k$ for every simple right $A$ -module $V$ . Let $e_{1}, \dots, e_{r}$ be primitive idempotents representing the principal indecomposables. Then

c_{i j} = \dim_{k} e_{i} A e_{j} (1 \leq i, j \leq r) .

(25.E)

Over a splitting field the Cartan invariants are literally the dimensions of the corners of $A$ .

Consequently the left Cartan matrix of such an algebra is the transpose of the right one.

Proof

For any idempotent $e$ and any finite-dimensional right $A$ -module $M$ we have ${Hom}_{A} (e A, M) ≅ M e$ , so $\dim_{k} {Hom}_{A} (e A, M) = \dim_{k} M e$ . The functor ${Hom}_{A} (e A, -)$ is exact because $e A$ is projective, so $\dim_{k} M e$ is additive along a composition series of $M$ . For a simple module $S$ , ${Hom}_{A} (e_{i} A, S)$ vanishes unless $S ≅ V_{i}$ , in which case it is ${End}_{A} (V_{i}) = k$ , of dimension $1$ by the splitting hypothesis. Hence $\dim_{k} M e_{i}$ equals the multiplicity of $V_{i}$ in $M$ . Taking $M = e_{i} A$ and the idempotent $e_{j}$ gives $c_{i j} = \dim_{k} e_{i} A e_{j}$ . The left-hand statement is the same computation for $A^{op}$ , whose corners satisfy $\dim_{k} e_{j} A e_{i} = c_{j i}$ .

Theorem—Brauer–Nesbitt–Nakayama, for group algebras

Let $(K, 𝒪, k)$ be a $p$ -modular system that is a splitting system for the finite group $G$ , with $char k = p$ dividing $| G |$ , and let $D$ be the decomposition matrix of $G$ . Then the Cartan matrix of $k G$ satisfies $C = D^{T} D$ . In particular $C$ is symmetric and positive definite, and $\det C$ is a power of $p$ .

Remark—Scope

Symmetry is special to symmetric algebras such as group algebras; it fails for $T_{n} (k)$ with $n \geq 2$ . If $char k$ does not divide $| G |$ then $k G$ is semisimple by Maschke's theorem and $C$ is the identity matrix. Lam records these facts without proof, noting that they belong to modular representation theory rather than to general ring theory.

Worked Example

Upper triangular matrices

Let $k$ be a division ring, $R = T_{4} (k)$ , $e_{i} = E_{i i}$ . Then $P_{i} = e_{i} R$ is the $i$ -th row, of $k$ -dimension $5 - i$ , and $e_{i} J ≅ e_{i + 1} R$ . The composition factors of $P_{i}$ are $V_{i}, V_{i + 1}, \dots, V_{4}$ , each once, giving

C = (\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}), \det C = 1, row sums 4, 3, 2, 1 .

(E.1)

Ones on and above the diagonal. The row sums are the lengths of the principal indecomposables, and $\sum_{i} (5 - i) = 10 = \dim_{k} T_{4} (k)$ .

The left Cartan matrix is the transpose — ones on and below the diagonal — reflecting that $R e_{i}$ is the $i$ -th column, of length $i$ . All entries off the block structure are irrelevant here: $e_{1} R e_{4} = k E_{14} \neq 0$ , so all four idempotents are linked, $R$ is indecomposable, and $C$ is a single indecomposable block, even though $\bar{R} ≅ k \times k \times k \times k$ .

The first-row algebra

Let $k$ be a division ring and let $R \subseteq M_{n} (k)$ be spanned by the diagonal matrix units together with the first row:

R = \sum_{i = 1}^{n} k E_{i i} + \sum_{j = 2}^{n} k E_{1 j}, \dim_{k} R = 2 n - 1, J = rad R = \sum_{j = 2}^{n} k E_{1 j}, J^{2} = 0 .

(E.2)

With $e_{i} = E_{i i}$ one finds $e_{i} R = k E_{i i} ≅ V_{i}$ for $i \geq 2$ , so those principal indecomposables are simple, while $e_{1} R = k E_{11} \oplus ⨁_{j \geq 2} k E_{1 j}$ has $e_{1} J ≅ V_{2} \oplus \dots \oplus V_{n}$ , a semisimple module. Hence $e_{1} R$ has length $n$ with factors $V_{1}, V_{2}, \dots, V_{n}$ each once, and

C = (\begin{matrix} 1 & 1 & \dots & 1 \\ 0 & 1 & 0 \\ ⋮ & ⋱ \\ 0 & 0 & 1 \end{matrix}), \det C = 1 .

(E.3)

First row all ones, every other row a unit vector. The left Cartan matrix has first column all ones — again the transpose.

Since $e_{1} R e_{i} = k E_{1 i} \neq 0$ for every $i \geq 2$ , all idempotents are linked and $R$ is indecomposable; the principal indecomposables are pairwise non-isomorphic, so $R$ is also basic. Every submodule of $e_{1} R$ is projective — for $i \geq 2$ this is trivial and for $i = 1$ it follows from semisimplicity of $e_{1} J$ — so $R$ is right hereditary.

A group algebra

Take $G = S_{3}$ and $k$ a splitting field of characteristic $3$ . The $3$ -regular classes are those of $1$ and of a transposition, so there are two simple $k G$ -modules, the trivial one and the sign, both one-dimensional. The ordinary irreducible characters have degrees $1, 1, 2$ , and the two-dimensional one reduces modulo $3$ with factors trivial and sign, so

D = (\begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{matrix}), C = D^{T} D = (\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}), \det C = 3 .

(E.4)

Symmetric, with determinant a power of $3$ , as Brauer–Nesbitt–Nakayama requires. Both principal indecomposables have dimension $3$ , and $2 \cdot 3 = 6 = | G |$ checks the dimension count.

Comparison and Classification

Cartan matrices of standard rings
Ring	$r$	Cartan matrix $C$	$\det C$
Semisimple ring	number of components	identity	$1$
$ℤ / p^{m} ℤ$	1	$(m)$	$m$
$k [x] / (x^{m})$	1	$(m)$	$m$
$T_{n} (k)$ , $k$ a division ring	$n$	ones on and above the diagonal	$1$
First-row algebra $(E . 2)$	$n$	first row ones, else identity	$1$
$k C_{p}$ , $char k = p$	1	$(p)$	$p$
$k S_{3}$ , $char k = 3$ , split	2	$(\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix})$	$3$
$M_{n} (A)$ for $A$ right artinian	same as $A$	same as $A$	same as $A$

Which hypotheses each property of $C$ needs
	Semiperfect	Right artinian	Finite-dimensional split	Group algebra, split
$C$ is defined	no	yes	yes	yes
$c_{i i} \geq 1$ and row sums are lengths	no	yes	yes	yes
Block diagonal along blocks	no	yes	yes	yes
$c_{i j} = \dim_{k} e_{i} A e_{j}$	no	no	yes	yes
$C$ symmetric	no	no	no	yes
$\det C$ a power of the characteristic	no	no	no	yes

Which hypotheses each property of $C$ needs

Relationship Map

The Cartan matrix sits between two lists and two Grothendieck groups.

K_{0} R

[e_{i} R]

Cartan map

c

G_{0} R

[V_{j}]

coker of order

| \det C |

Data determined by $C$
- Reads off directly
  - composition lengths of the principal indecomposables (row sums)
  - which simple modules meet which projectives (support)
  - the partition into blocks (indecomposable diagonal blocks)
- Reads off with extra input
  - dimensions, given the dimensions of the simple modules
  - the order of the cokernel of the Cartan map, via $\det C$
  - the decomposition matrix, when a $p$ -modular system is available
- Does not determine
  - the ring up to isomorphism
  - the module category — non-isomorphic algebras share Cartan matrices
  - the submodule lattice of any $e_{i} R$

Passing to a basic ring or to $M_{n} (R)$ leaves $C$ unchanged up to relabelling, since composition multiplicities of principal indecomposables are preserved by any equivalence of module categories.

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Modular representation theory

Brauer theory

Cartan invariants, decomposition numbers and defect groups organise the representation theory of a finite group in characteristic dividing the order; $\det C$ detects the defect of a block.

Algebraic K-theory

The Cartan homomorphism

$C$ is the matrix of $K_{0} R \to G_{0} R$ . Its invertibility over $ℤ$ is equivalent to that map being an isomorphism, which holds for rings of finite global dimension such as $T_{n} (k)$ .

Quiver representations

Reading off the algebra

For a basic algebra given by a quiver with relations, the Cartan invariants count paths modulo relations from vertex $i$ to vertex $j$ , so $C$ is a combinatorial invariant of the presentation.

Computational chemistry and physics

Symmetry-adapted bases

Where finite group representations are used to block-diagonalise operators, characteristic-dividing cases require modular data; Cartan and decomposition matrices are the bookkeeping for how ordinary representations degenerate.

As with the rest of this section, the honest description is internal: the Cartan matrix is the standard compression of the composition data of an artinian ring, and it is the form in which that data is tabulated, transmitted and compared.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

This collection

c_{i j}

= multiplicity of

V_{j}

e_{i} R

; rows indexed by projectives

Common variantThe transpose, with

c_{i j}

= multiplicity of

V_{i}

P_{j}

; check before comparing

Bracket notation

[P_{i} : V_{j}]

[e_{i} R : V_{j}]

for the multiplicity

SideRight Cartan matrix unless stated; the left one is a different invariant in general

Group algebras

C = D^{T} D

with

D

the decomposition matrix, over a splitting

p

-modular system

SymbolsISO 80000-2 conventions for matrices; upright

\det

and

\dim

GAPBrauer tables and DecompositionMatrix for group algebras

Other systemsMagma and Sage provide radical, simple modules and Cartan matrices for finite-dimensional algebras

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

Radical and simplesCompute

rad A

and split

A / rad A

; this produces

r

and the simple modules

V_{1}, \dots, V_{r}

IdempotentsLift primitive idempotents from the semisimple quotient to obtain

e_{1}, \dots, e_{r}

representing the principal indecomposables.

CornersOver a splitting field, set

c_{i j} = \dim_{k} e_{i} A e_{j}

— a rank computation on the multiplication table. Otherwise compute composition series of each

e_{i} A

Order and verifySort the indices by block, check

c_{i i} \geq 1

and that the row sums equal the lengths, and confirm the diagonal blocks are indecomposable.

The dominant cost is the radical computation and the splitting of the semisimple quotient; forming $\dim_{k} e_{i} A e_{j}$ afterwards is $O (r^{2})$ rank computations on subspaces of $A$ .
Without a splitting field one must work with ${End}_{A} (V_{j})$ : the multiplicity is $\dim_{k} M e_{j}$ divided by $\dim_{k} {End}_{A} (V_{j})$ , so the division ring degrees must be computed first.
For group algebras the Cartan matrix is usually obtained from the decomposition matrix via $C = D^{T} D$ ; computing $D$ is the hard step and is where the MeatAxe and Brauer character machinery are used.
Computer algebra systems expose these as primitives: GAP's character table library supplies Brauer tables and decomposition matrices, and general algebra packages provide radical, simple-module and Cartan-matrix routines for finite-dimensional algebras.

Failure Modes and Common Mistakes

Do not read $c_{i j}$ as a dimension unless the base field is a splitting field; in general it is a multiplicity, and dimensions carry an extra factor $\dim_{k} End (V_{j})$ .
Do not conclude that two algebras with equal Cartan matrices are isomorphic or even Morita equivalent; $C$ is a coarse invariant.
Do not confuse the Cartan matrix with the decomposition matrix. They have different shapes: $D$ has one row per ordinary irreducible character, $C$ is square.
Do not expect $\det C = 1$ outside rings of finite global dimension; $ℤ / p^{m} ℤ$ already has $\det C = m$ .

Quick Reference

Hypothesis

R

right artinian;

e_{1} R, \dots, e_{r} R

the principal indecomposables;

V_{i} = e_{i} R / e_{i} J

Definition

c_{i j}

= number of composition factors of

e_{i} R

isomorphic to

V_{j}

Diagonal

c_{i i} \geq 1

, since

V_{i}

is the top of

e_{i} R

Rows

\sum_{j} c_{i j}

= composition length of

e_{i} R

Support

c_{i j} \neq 0 \Leftrightarrow e_{i} R e_{j} \neq 0

Blocks

C = diag (C_{1}, \dots, C_{s})

, one indecomposable block per block of

R

Split algebras

c_{i j} = \dim_{k} e_{i} A e_{j}

; the left Cartan matrix is

C^{T}

Group algebras

C = D^{T} D

; symmetric, with

\det C

a power of

p

Statement locator
Fact	Statement	Reference
Definition	$C = (c_{i j}) \in M_{r} (ℤ)$	§25 (p. 376)
Factor criterion	$V_{j}$ a factor of $M$ iff $M e_{j} \neq 0$	(21.19)
Block form	$C = diag (C_{1}, \dots, C_{s})$	§25 (p. 376)
Cartan map	$C$ is the matrix of $K_{0} R \to G_{0} R$	§25 (p. 377)
Group algebra case	$C$ symmetric, $\det C$ a power of $char k$	§25 (p. 377)
Triangular example	ones on and above the diagonal	§25 (pp. 377–378)
Corner dimensions	$c_{i j} = \dim_{k} e_{i} A e_{j}$ over a splitting field	§25, Exercises 1–2

Frequently Asked Questions

Why is the Cartan matrix defined only for artinian rings?

Because it counts composition factors, and a module needs a composition series for that count to exist and be unique. Over a right artinian ring every finitely generated right module has one. A semiperfect ring that is not right artinian can have principal indecomposables of infinite length, in which case the invariants are simply not defined.

Are the left and right Cartan matrices transposes of each other?

For a finite-dimensional algebra over a splitting field, yes: both count corner dimensions, and $\dim_{k} e_{i} A e_{j}$ appears as $c_{i j}$ on the right and as the $(j, i)$ entry on the left. In general the relationship is more delicate, and if the ring is artinian on only one side the other matrix may not exist.

What does

\det C = 1

tell me?

That the Cartan map $K_{0} R \to G_{0} R$ is an isomorphism, so every finitely generated module has a well-defined class expressible in terms of projectives. This holds for rings of finite global dimension, such as $T_{n} (k)$ and the first-row algebra, and fails for $ℤ / p^{m} ℤ$ with $m \geq 2$ .

Can two non-isomorphic algebras have the same Cartan matrix?

Easily. The Cartan matrix records multiplicities but not extensions, so algebras with the same composition data and different module structure share it. It is a useful invariant precisely because it is cheap, not because it is complete.

How does the Cartan matrix change under Morita equivalence?

Not at all, up to simultaneous permutation of rows and columns. In particular $R$ , its basic ring, and $M_{n} (R)$ all have the same Cartan matrix, since an equivalence matches principal indecomposables with principal indecomposables and simple modules with simple modules.

What is the relationship between the Cartan matrix and the quiver of an algebra?

For a basic algebra over an algebraically closed field, the number of arrows from $i$ to $j$ in the quiver is $\dim {Ext}^{1} (V_{i}, V_{j})$ , while $c_{i j}$ counts all occurrences of $V_{j}$ in $e_{i} A$ , including those arising from longer paths. The quiver is the finer invariant; the Cartan matrix is a shadow of it.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §25 (pp. 376–380).
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, §16 and §18 (Cartan and decomposition matrices).
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer-Verlag, 1977, Part III.
R. Brauer and C. Nesbitt, “On the modular characters of groups”, Annals of Mathematics 42 (1941), 556–590.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §27–§32.
I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1, Cambridge University Press, 2006, Chapter III.

AI Suggested Questions

Compute the Cartan matrix of the group algebra of the alternating group on four letters in characteristic $2$ .
Show that a right artinian ring of finite global dimension has $\det C = \pm 1$ , and give a proof or a counterexample for the converse.
Work out the Cartan matrix of a path algebra $k Q$ modulo the square of its arrow ideal.
How do Cartan invariants behave under tensor products of finite-dimensional algebras over a field?
Explain the proof that $C = D^{T} D$ for group algebras over a splitting $p$ -modular system.
Which symmetric positive definite integer matrices with positive entries actually occur as Cartan matrices of finite-dimensional algebras?
Describe how the Cartan matrix of a block of $k G$ constrains its defect group.

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Testing for a composition factor

Why C splits along blocks

The K-theoretic reading

06Key Results

07Worked Example

Upper triangular matrices

The first-row algebra

A group algebra

08Comparison and Classification

09Relationship Map

10Applications and Industry Use

Brauer theory

The Cartan homomorphism

Reading off the algebra

Symmetry-adapted bases

11Standards and Notation

12Computational Notes

13Failure Modes and Common Mistakes

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why $C$ splits along blocks

Key Results

Worked Example

Comparison and Classification

Relationship Map

Applications and Industry Use

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions