Blocks of Finite-Dimensional Algebras and Group Algebras

Q: Why does the centre act by scalars on a simple module?

A central element acts as an R-module endomorphism, and for a simple finite-dimensional module over an algebraically closed field Schur's Lemma makes the endomorphism ring equal to k. So the action is multiplication by an element of k. Over a non-closed field the endomorphism ring can be a larger division algebra and the conclusion fails.

Q: Is the number of blocks the same as the number of irreducible characters?

Only in the semisimple case. In characteristic p dividing the group order, several simple modules typically share a block: kS_3 in characteristic 3 has two simple modules and one block. The general inequality is that blocks are at most as many as simple modules.

Q: How do I compute the blocks of a group algebra in practice?

Work in Z(kG) with the class sums as a basis, compute its radical, factor the semisimple quotient into a product of copies of k, and lift the resulting idempotents back through the radical. Everything happens in dimension the number of conjugacy classes rather than the group order.

Q: What is the principal block?

The block containing the trivial module. Its central character sends a class sum to the class size, reduced into k. It is the block that always exists, and Brauer's first main theorem relates it to the principal block of the normaliser of a Sylow subgroup.

Q: Does (22.7) need the algebra to be split, or algebraically closed?

Split is enough: all that is used is End_R(S) = k for every simple S. Algebraic closure is the convenient sufficient condition, and for group algebras a large enough finite field — one containing the relevant roots of unity — is already a splitting field by Brauer's theorem.

Q: Can two blocks have the same central character on a subalgebra of the centre?

Yes, and this is a real hazard when working with a proper subalgebra such as the span of a few class sums. The characterisation in (22.7) requires the full centre; restricting to part of it merges blocks.

Executive Summary

For a finite-dimensional algebra $R$ over an algebraically closed field $k$ , blocks can be read off from the centre alone. Each primitive idempotent $e$ gives a simple module $e R / e rad R$ , on which every central element acts as a scalar by Schur's Lemma. The resulting map $λ_{e} : Z (R) \to k$ is a $k$ -algebra homomorphism, and $(22.7)$ says $λ_{e}$ is a complete invariant of the block of $e$ .

The practical consequence is a count: the number of blocks equals the number of $k$ -algebra homomorphisms $Z (R) \to k$ , equivalently $\dim_{k} Z (R) / rad Z (R)$ . For a group algebra the centre has an explicit basis of class sums, which turns block theory into linear algebra of size the number of conjugacy classes rather than $| G |$ .

$λ_{e} : Z (R) \to k$ Central character

$λ_{e} = λ_{e'}$ Same block

$\dim_{k} Z / rad Z$ Number of blocks

$#$ classes

\dim_{k} Z (k G)

Overview

Suppose $1 = c_{1} + \dots + c_{r}$ is a decomposition into orthogonal centrally primitive idempotents — always available for a finite-dimensional algebra, since its ideals satisfy both chain conditions. Then $R$ and its centre $C = Z (R)$ decompose in parallel:

R = R_{1} \oplus \dots \oplus R_{r}, C = C_{1} \oplus \dots \oplus C_{r}, R_{i} = c_{i} R, C_{i} = c_{i} C = Z (R_{i}) .

(22.7a)

The centre of a product is the product of the centres, so the block structure of $C$ mirrors that of $R$ exactly.

Each $C_{i}$ is a finite-dimensional commutative $k$ -algebra with no idempotents other than $0$ and $c_{i}$ , hence local; over an algebraically closed field its residue field is $k$ itself, so $C_{i}$ admits exactly one $k$ -algebra homomorphism to $k$ . Counting blocks is therefore counting homomorphisms $C \to k$ , and the whole of $(22.7)$ is the identification of those homomorphisms with the way central elements act on simple modules.

Algebraic closure is used exactly once, in Schur's Lemma, to know that ${End}_{R}$ of a simple module is $k$ rather than a larger division algebra. Without it the theorem must be restated with central characters valued in field extensions of $k$ , which is why split algebras are the natural setting.

Learning Objectives

Construct the central character $λ_{e}$ of a primitive idempotent and verify it is a $k$ -algebra homomorphism.
Prove that each block $C_{i}$ of the centre is a local $k$ -algebra with residue field $k$ .
Prove $(22.7)$ : $λ_{e} = λ_{e'}$ if and only if $e$ and $e'$ lie in the same block.
Deduce that the number of blocks equals $\dim_{k} C / rad C$ .
Use class sums to compute the centre of a group algebra and its blocks.
Contrast the block structure of the group algebra of the symmetric group on three letters in characteristics 0 and 3.

Definitions

Definition(22.7b)Central character of a primitive idempotent

Let $R$ be a finite-dimensional algebra over an algebraically closed field $k$ , let $J = rad R$ and let $e \in R$ be a primitive idempotent. The module $M_{e} = e R / e J$ is a simple right $R$ -module. For $c \in C = Z (R)$ the map $M_{e} \to M_{e}$ , $x \mapsto x c$ , is an $R$ -module endomorphism, hence multiplication by a scalar $λ_{e} (c) \in k$ . The map $λ_{e} : C \to k$ is the central character of $e$ .

$C = Z (R)$: The centre. A finite-dimensional commutative $k$ -algebra, hence a finite product of local $k$ -algebras.
$M_{e} = e R / e rad R$: The simple top of the indecomposable projective $e R$ . Every simple right $R$ -module is of this form.
$λ_{e}$: The scalar action of the centre on $M_{e}$ ; a $k$ -algebra homomorphism $C \to k$ , necessarily surjective and with kernel a maximal ideal of $C$ .
Class sum $\hat{K}$: For a conjugacy class $K$ of a finite group $G$ , the element $\sum_{g \in K} g$ of $k G$ . These form a $k$ -basis of $Z (k G)$ .
Split algebra: ${End}_{R} (S) = k$ for every simple $R$ -module $S$ . Automatic when $k$ is algebraically closed.

Throughout, $k$ is algebraically closed and $\dim_{k} R < \infty$ . Modules are right modules; the left-handed statements are identical.

Core Concepts

Why the centre acts by scalars

If $c \in Z (R)$ and $M$ is any right $R$ -module, then $x \mapsto x c$ commutes with the action of $R$ , so it is an endomorphism of $M$ . When $M$ is simple and finite-dimensional over an algebraically closed $k$ , Schur's Lemma plus algebraic closure give ${End}_{R} (M) = k$ : the endomorphism ring is a finite-dimensional division algebra over $k$ , hence $k$ itself. So the action of $c$ is multiplication by a scalar.

c \in Z (R)

endomorphism of

M

{End}_{R} (M) = k

scalar

λ (c)

Blocks of the centre are local

The idempotents of $C_{i}$ are precisely the central idempotents of $R$ lying in $c_{i} R$ , and there are only $0$ and $c_{i}$ because $c_{i}$ is centrally primitive. A finite-dimensional algebra with no nontrivial idempotents is local, so $C_{i}$ is a local commutative $k$ -algebra with $C_{i} / rad C_{i}$ a finite field extension of $k$ , hence equal to $k$ . Consequently $C_{i}$ has exactly one $k$ -algebra homomorphism onto $k$ , namely reduction modulo its radical.

C / rad C ≅ {\underset{⏟}{k \times \dots \times k}}_{r}, r = \dim_{k} C / rad C = # {blocks} .

(22.7c)

Blocks correspond to maximal ideals of the centre.

The centre of a group algebra

For a finite group $G$ , an element of $k G$ is central exactly when its coefficient function is constant on conjugacy classes, so the class sums ${\hat{K}}_{1}, \dots, {\hat{K}}_{s}$ form a $k$ -basis of $Z (k G)$ and $\dim_{k} Z (k G) = s$ , the number of conjugacy classes, in every characteristic. Multiplication is governed by the structure constants ${\hat{K}}_{i} {\hat{K}}_{j} = \sum_{m} a_{i j m} {\hat{K}}_{m}$ with $a_{i j m}$ non-negative integers reduced into $k$ .

Key Results

Theorem(22.7)Blocks are the fibres of the central character

Let $R$ be a finite-dimensional algebra over an algebraically closed field $k$ , with block decomposition $R = R_{1} \oplus \dots \oplus R_{r}$ and corresponding decomposition $C = C_{1} \oplus \dots \oplus C_{r}$ of $C = Z (R)$ , where $C_{i} = Z (R_{i})$ . Let $E$ be the set of primitive idempotents of $R$ . Then:

for each $e \in E$ , every $c \in C$ acts on the simple module $e R / e rad R$ as multiplication by a scalar $λ_{e} (c) \in k$ , and $λ_{e} : C \to k$ is a $k$ -algebra homomorphism;
$e, e' \in E$ lie in the same block if and only if $λ_{e} = λ_{e'}$ ;
every $k$ -algebra homomorphism $C \to k$ equals $λ_{e}$ for some $e \in E$ .

Proof

(1). $R$ is finite-dimensional, hence semiperfect, so a primitive idempotent $e$ is local and $M_{e} = e R / e rad R$ is a nonzero simple module. Right multiplication by $c \in C$ is an $R$ -endomorphism of $M_{e}$ , and ${End}_{R} (M_{e}) = k$ by Schur's Lemma and algebraic closure. So the action is by a scalar $λ_{e} (c)$ . Additivity, multiplicativity and $λ_{e} (1) = 1$ are immediate from the corresponding properties of the action, so $λ_{e}$ is a $k$ -algebra homomorphism.

(2). By $(22.4)$ each $e \in E$ lies in exactly one block, say $e \in R_{i}$ , so $e c_{i} = e$ and $e c_{j} = 0$ for $j \neq i$ . Then $M_{e} c_{j} = 0$ for $j \neq i$ , giving $λ_{e} (c_{j}) = 0$ and $λ_{e} (c_{i}) = 1$ . Since $C_{j} = c_{j} C$ , this forces $λ_{e} |_{C_{j}} = 0$ for $j \neq i$ .

On $C_{i}$ , the map $λ_{e} |_{C_{i}}$ is a $k$ -algebra homomorphism onto $k$ ; as shown above, $C_{i}$ is local with $C_{i} / rad C_{i} = k$ , so there is exactly one such homomorphism, call it $λ^{(i)}$ . Hence $λ_{e}$ is determined by the block index $i$ : it is $λ^{(i)}$ on $C_{i}$ and zero on the other summands. Conversely, $λ_{e} (c_{i}) = 1$ recovers $i$ from $λ_{e}$ , so distinct blocks give distinct characters. This is exactly (2).

(3). Let $λ : C \to k$ be any $k$ -algebra homomorphism. Each $λ (c_{i})$ is an idempotent of $k$ , hence $0$ or $1$ , the $λ (c_{i})$ are pairwise orthogonal, and $\sum_{i} λ (c_{i}) = λ (1) = 1$ . So exactly one index $i$ has $λ (c_{i}) = 1$ , and $λ$ vanishes on every $C_{j}$ with $j \neq i$ . Its restriction to $C_{i}$ is a $k$ -algebra homomorphism onto $k$ , hence equals $λ^{(i)}$ by uniqueness. Thus $λ = λ_{e}$ for any $e \in E$ lying in $R_{i}$ — and such $e$ exists, since $1 = \sum c_{i}$ can be refined into orthogonal primitive idempotents, at least one of which lies in $R_{i}$ .

Corollary(22.7d)Counting blocks

With $R$ and $k$ as above, the number of blocks of $R$ equals the number of maximal ideals of $C = Z (R)$ , equals the number of $k$ -algebra homomorphisms $C \to k$ , equals $\dim_{k} C / rad C$ . In particular the number of blocks of $R$ depends only on the commutative algebra $Z (R)$ .

Proof

$C = C_{1} \oplus \dots \oplus C_{r}$ with each $C_{i}$ local, so the maximal ideals of $C$ are the $r$ ideals $rad C_{i} \oplus ⨁_{j \neq i} C_{j}$ , and $C / rad C ≅ k^{r}$ . A $k$ -algebra homomorphism onto $k$ has a maximal ideal as kernel and is determined by it, so the three counts agree; $(22.7) (3)$ identifies them with the set of blocks.

Corollary(22.7e)Group algebras

Let $G$ be a finite group and $k$ an algebraically closed field. The blocks of $k G$ correspond bijectively to the $k$ -algebra homomorphisms $Z (k G) \to k$ , where $Z (k G)$ has the class sums as a basis. If $char k ∤ | G |$ then $k G$ is semisimple by Maschke's Theorem, $Z (k G) ≅ k^{s}$ with $s$ the number of conjugacy classes, and the blocks are the $s$ matrix components — one per irreducible character. If $char k = p$ divides $| G |$ , then $Z (k G)$ has a nonzero radical and the number of blocks is generally smaller than $s$ .

Remark(22.7f)The principal block

The trivial module $k$ is simple, and a class sum $\hat{K}$ acts on it by $| K | \cdot 1_{k}$ . So the corresponding central character is $λ_{0} (\hat{K}) = | K |$ in $k$ , and the block it labels is the principal block. Identifying which other simple modules share this character is the first computation performed in modular representation theory.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Push the problem into the centre

Whenever an invariant is constant on a block, it factors through $Z (R)$ . Since $Z (R)$ is commutative and usually far smaller than $R$ , the whole classification becomes a question about a commutative artinian ring.

Move 2

Local algebra, unique character

A finite-dimensional local $k$ -algebra with residue field $k$ has exactly one $k$ -algebra map to $k$ . This turns 'no nontrivial idempotents' into 'exactly one character' and is the pivot of the proof.

Move 3

Test with the block idempotents

Evaluating an unknown homomorphism on $c_{1}, \dots, c_{r}$ forces a $0 / 1$ pattern with a single $1$ . This locates the block and is how surjectivity in part (3) is obtained without any construction.

The theorem is best remembered as a change of category: from modules over a possibly wild noncommutative algebra to points of the finite scheme $Spec Z (R)$ . The blocks are the connected components of that scheme, and the central characters are its $k$ -points.

Worked Example

The symmetric group on three letters

Let $G = S_{3}$ with conjugacy classes ${1}$ , the three transpositions, and the two $3$ -cycles. Write $T$ for the sum of the transpositions and $A = a + a^{2}$ for the sum of the $3$ -cycles, where $a = (1 2 3)$ . Then $Z (k G)$ has basis ${1, T, A}$ over any field $k$ , so $\dim_{k} Z (k G) = 3$ .

Structure constants

Multiplying out: among the nine products of two transpositions, three give the identity and six give $3$ -cycles, each $3$ -cycle arising three times. Each transposition times each $3$ -cycle is again a transposition, and for fixed $t$ the elements $t a$ and $t a^{2}$ are the two transpositions other than $t$ . Hence, over $ℤ$ ,

T^{2} = 3 + 3 A, A^{2} = 2 + A, T A = A T = 2 T .

(E.1)

Structure constants of $Z (ℤ S_{3})$ in the class-sum basis.

Characteristic zero: three blocks

If $char k = 0$ (or any characteristic prime to $6$ ), Maschke's Theorem gives $k S_{3} ≅ k \times k \times M_{2} (k)$ and $Z (k S_{3}) ≅ k \times k \times k$ . There are three $k$ -algebra homomorphisms to $k$ , hence three blocks, one for each irreducible character — trivial, sign, and the two-dimensional one. Their central characters are computed from $ω_{χ} (\hat{K}) = | K | χ (x_{K}) / χ (1)$ ; for instance the two-dimensional character has $ω (T) = 3 \cdot 0 / 2 = 0$ and $ω (A) = 2 \cdot (- 1) / 2 = - 1$ .

Characteristic three: one block

Now let $char k = 3$ with $k$ algebraically closed. Reducing $(E . 1)$ : $T^{2} = 0$ , $A^{2} = A + 2$ , $T A = 2 T$ . Put $x = A + 1 = 1 + a + a^{2}$ . Then

x^{2} = A^{2} + 2 A + 1 = (A + 2) + 2 A + 1 = 3 A + 3 = 0, T x = T A + T = 2 T + T = 3 T = 0 .

(E.2)

So $Z (k S_{3}) = k \cdot 1 \oplus k x \oplus k T$ with $x^{2} = x T = T^{2} = 0$ : a local $k$ -algebra whose radical is $k x + k T$ and has square zero. There is exactly one $k$ -algebra homomorphism to $k$ , so by $(22.7 d)$ the algebra $k S_{3}$ has one block in characteristic $3$ — it is indecomposable, even though it is not simple.

This is consistent with the module count. The augmentation ideal of the normal Sylow $3$ -subgroup $N = ⟨ a ⟩$ generates a nilpotent ideal $I (N) k G$ with quotient $k [G / N] = k C_{2}$ , which is semisimple because $3 ∤ 2$ . So $rad (k S_{3}) = I (N) k S_{3}$ and $k S_{3}$ has exactly two simple modules, both one-dimensional: the trivial and the sign representation. Both must have the same central character, since only one exists — and indeed both lie in the single block.

Comparison and Classification

Three numbers attached to a finite-dimensional algebra over an algebraically closed field
Algebra	Simple modules	Blocks	$\dim_{k} Z (R)$
$M_{n} (k)$	$1$	$1$	$1$
$T_{n} (k)$ , $n \geq 2$	$n$	$1$	$1$
$k [x] / (x^{m})$	$1$	$1$	$m$
$k S_{3}$ , $char k = 0$	$3$	$3$	$3$
$k S_{3}$ , $char k = 3$	$2$	$1$	$3$
$k G$ , $char k ∤ \| G \|$	$s$ classes	$s$	$s$
$k P$ , $P$ a $p$ -group, $char k = p$	$1$	$1$	$#$ classes of $P$

The three columns are independent. Only in the semisimple case do they all coincide; the last two rows show that the centre can be large while the block count is one.

Relationship Map

Block of $R$ — for $R$ finite-dimensional over algebraically closed $k$
- is labelled by
  - a centrally primitive idempotent $c_{i}$
  - a maximal ideal of $Z (R)$
  - a $k$ -algebra homomorphism $Z (R) \to k$
  - a connected component of the linkage graph
- contains
  - one or more isomorphism classes of simple modules
  - the corresponding indecomposable projectives
  - all extensions between them
- is detected by
  - $λ_{e} (c_{i}) = 1$
  - common composition factors of the projectives

The first and last labels join the two descriptions of a block given in this collection: The Block Decomposition of a Ring builds blocks from linkage of primitive idempotents, while $(22.7)$ builds them from the centre. Both partitions of the simple modules coincide, and each is easier to compute in different circumstances.

Simple module

M_{e}

Central character

λ_{e}

Maximal ideal of

Z (R)

Block

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's partition of characters

Working over a $p$ -modular system, the central character of an ordinary irreducible character reduces to a $k$ -algebra map $Z (k G) \to k$ , and characters are declared to lie in the same $p$ -block when their reductions agree. All of Brauer's block theory — defect groups, the three main theorems — is built on this partition.

Computational group theory

Block distribution from the character table

Given the ordinary character table and the class sizes, block distribution in characteristic $p$ is a congruence computation on the numbers $| K | χ (x_{K}) / χ (1)$ . This is why systems can report $p$ -blocks without ever constructing $k G$ .

Quantum groups and Lie theory

Central characters elsewhere

The same device — act by the centre, get a scalar, partition by the resulting character — organises category $𝒪$ for a semisimple Lie algebra via the Harish-Chandra homomorphism, and the representation theory of quantum groups at roots of unity.

Symbolic computation

Small linear algebra

Block computations run in $Z (R)$ , of dimension the number of conjugacy classes, instead of $R$ , of dimension $| G |$ . For a group of order $10^{5}$ with $60$ classes this is a $60$ -dimensional problem.

Honest summary: this theorem is the reduction step that makes modular representation theory computable at all. It is used constantly and studied rarely.

Computational Notes

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

Build $Z (R)$ : for a group algebra, take the class sums; for a general algebra of dimension $n$ , solve the linear system $x b_{i} = b_{i} x$ at cost $O (n^{4})$ naively and much less with structure.
Compute $rad Z (R)$ using the trace form in characteristic $0$ , or the Friedl–Rónyai method in characteristic $p$ .
Factor $Z (R) / rad Z (R) ≅ k^{r}$ into its primitive idempotents; over a finite field this is polynomial factorisation.
Lift each primitive idempotent through the nilpotent ideal $rad Z (R)$ by the Newton step $e \mapsto 3 e^{2} - 2 e^{3}$ , doubling accuracy each round.
The lifted elements are the block idempotents $c_{1}, \dots, c_{r}$ of $R$ .

Note that steps 2 to 5 all take place in a commutative algebra of dimension the number of conjugacy classes; the noncommutative structure of $R$ is never touched. The cost is therefore governed by the centre and not by $\dim_{k} R$ .

Failure Modes and Common Mistakes

Do not compute $λ_{e}$ on $e R$ itself; the projective module is not simple in general, and the centre does not act on it by scalars.
Do not assume the central character determines the simple module: it determines only the block, and a block can contain many simple modules.
Do not transfer the group-algebra formula $ω_{χ} (\hat{K}) = | K | χ (x_{K}) / χ (1)$ into characteristic $p$ without a $p$ -modular system; the division by $χ (1)$ needs the characteristic-zero setting first, and reduction comes afterwards.

Historical Notes and Lessons Learned

1896Frobenius introduces charactersGroup characters appear as factorisations of the group determinant; the central characters $ω_{χ}$ are implicit in the algebra of class sums from the start.
1907–1908Wedderburn's structure theoryFinite-dimensional algebras over a field are analysed through their radical and semisimple quotient, giving the semisimple case of the block picture.
1935–1941Brauer and Nesbitt define blocksModular representation theory is founded, and ordinary characters are partitioned into $p$ -blocks by congruences on their central characters.
1940s–1950sDefect groups and the main theoremsBrauer attaches a $p$ -subgroup to each block and proves the correspondence theorems relating blocks of $G$ to blocks of local subgroups.
1990Broué's conjectureBroué proposes that a block with abelian defect group is derived equivalent to its Brauer correspondent, moving block theory into homological algebra where it remains a central open problem.

The methodological lesson is the same one that produced the Jacobson radical: an invariant defined by action on modules — here, the scalar by which the centre acts — outperforms one defined by internal structure. Central characters are computable from the character table alone, long before anyone constructs the block algebras themselves.

Quick Reference

Setting

\dim_{k} R < \infty

k

algebraically closed,

C = Z (R)

Central character

c

acts on

e R / e rad R

λ_{e} (c) \in k

Main statement

(22.7)

: same block iff

λ_{e} = λ_{e'}

; all characters arise

Block count

r = \dim_{k} C / rad C

= number of maximal ideals of

C

Group algebra

Z (k G)

has the class sums as basis;

\dim = #

conjugacy classes

Principal block

λ_{0} (\hat{K}) = | K |

k

Semisimple caseblocks = simple components = irreducible characters

Warning

#

blocks

\leq #

simple modules, usually strict in characteristic

p

What each hypothesis buys
Hypothesis	Used for	Fails without it
$\dim_{k} R < \infty$	block decomposition exists; $e$ local; $C$ artinian	no finite block decomposition need exist
$k$ algebraically closed	${End}_{R} (M) = k$ ; $C_{i} / rad C_{i} = k$	central elements need not act by scalars in $k$
$e$ primitive	$e R / e rad R$ simple	the quotient is semisimple but not simple, and no single scalar exists
$c_{i}$ centrally primitive	$C_{i}$ local, unique character	several characters per factor

Frequently Asked Questions

Why does the centre act by scalars on a simple module?

A central element acts as an $R$ -module endomorphism, and for a simple finite-dimensional module over an algebraically closed field Schur's Lemma makes the endomorphism ring equal to $k$ . So the action is multiplication by an element of $k$ . Over a non-closed field the endomorphism ring can be a larger division algebra and the conclusion fails.

Is the number of blocks the same as the number of irreducible characters?

Only in the semisimple case. In characteristic $p$ dividing the group order, several simple modules typically share a block: $k S_{3}$ in characteristic $3$ has two simple modules and one block. The general inequality is that blocks are at most as many as simple modules.

How do I compute the blocks of a group algebra in practice?

Work in $Z (k G)$ with the class sums as a basis, compute its radical, factor the semisimple quotient into a product of copies of $k$ , and lift the resulting idempotents back through the radical. Everything happens in dimension the number of conjugacy classes rather than the group order.

What is the principal block?

The block containing the trivial module. Its central character sends a class sum to the class size, reduced into $k$ . It is the block that always exists, and Brauer's first main theorem relates it to the principal block of the normaliser of a Sylow subgroup.

Does

(22.7)

need the algebra to be split, or algebraically closed?

Split is enough: all that is used is ${End}_{R} (S) = k$ for every simple $S$ . Algebraic closure is the convenient sufficient condition, and for group algebras a large enough finite field — one containing the relevant roots of unity — is already a splitting field by Brauer's theorem.

Can two blocks have the same central character on a subalgebra of the centre?

Yes, and this is a real hazard when working with a proper subalgebra such as the span of a few class sums. The characterisation in $(22.7)$ requires the full centre; restricting to part of it merges blocks.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §22 (pp. 341–342), Theorem (22.7).
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, chapters on blocks, central characters and $p$ -modular systems.
J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Cambridge University Press, 1986.
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977, for the centre of a group ring and class sums.
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer-Verlag, 1977, §6 and §11, for central characters in characteristic zero.

AI Suggested Questions

Work out the blocks of $k A_{4}$ in characteristic $2$ and in characteristic $3$ .
How does the central character formula $ω_{χ} (\hat{K}) = | K | χ (x_{K}) / χ (1)$ reduce modulo a prime, and why is the value an algebraic integer?
State Brauer's first main theorem and explain how it refines the block partition produced by $(22.7)$ .
What is the defect group of a block, and why does a block of defect zero consist of a single matrix algebra?
Give an example of a finite-dimensional algebra over $ℝ$ where central elements do not act by scalars on a simple module.
How many blocks does $k G$ have when $G$ is a $p$ -group and $k$ has characteristic $p$ , and why?
Explain the analogy between central characters of a group algebra and the Harish-Chandra homomorphism for a semisimple Lie algebra.

Blocks of Algebras

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why the centre acts by scalars

Blocks of the centre are local

The centre of a group algebra

Key Results

Proof Techniques and Method

Push the problem into the centre

Local algebra, unique character

Test with the block idempotents

Worked Example

The symmetric group on three letters

Structure constants

Characteristic zero: three blocks

Characteristic three: one block

Comparison and Classification

Relationship Map

Applications and Industry Use

Brauer's partition of characters

Block distribution from the character table

Central characters elsewhere

Small linear algebra

Computational Notes

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why the centre acts by scalars

Blocks of the centre are local

The centre of a group algebra

06Key Results

07Proof Techniques and Method

Push the problem into the centre

Local algebra, unique character

Test with the block idempotents

08Worked Example

The symmetric group on three letters

Structure constants

Characteristic zero: three blocks

Characteristic three: one block

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Brauer's partition of characters

Block distribution from the character table

Central characters elsewhere

Small linear algebra

12Computational Notes

13Failure Modes and Common Mistakes

14Historical Notes and Lessons Learned

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

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Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

Comparison and Classification

Relationship Map

Applications and Industry Use

Computational Notes

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

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