Central Idempotents of a Group Algebra and Characters

Q: How is the corollary that the characteristic does not divide any degree consistent with the modular theory?

It is not a statement about arbitrary kG; it uses char k |G|. In that situation every n_i divides |G| in characteristic zero and the degrees carry over, so a prime not dividing |G| cannot divide any n_i. When char k does divide |G| the simple modules are different objects and their dimensions can be divisible by the characteristic.

Q: What replaces these formulas over the integers?

Nothing as clean. The e_i have denominators dividing |G|, so ZG typically has no idempotents at all besides 0 and 1 — that is exactly Theorem (8.26) on the *Integral Group Rings* page. The integral object that survives is Z(ZG) = _g ZC_g, a commutative ring finitely generated over Z, and its integrality is what powers the divisibility theorems.

Q: Is the class-sum basis multiplicative in a usable way?

Yes: C_g C_h = _x a_ghx C_x with a_ghx non-negative integers, the class multiplication coefficients. Applying _i turns this into a numerical identity among the quantities m_g_i(g)/n_i, which is precisely how the Burnside–Dixon algorithm recovers a character table from the class structure alone.

Q: Where does the factor n_i come from intuitively?

From the multiplicity of M_i in the regular module. The projection of kG onto its i-th Wedderburn component has rank n_i^2, and the coefficient of 1 in e_i is that rank divided by |G|. The factor n_i in the formula is the multiplicity, and the second factor n_i hiding in _i(1) is the dimension.

Executive Summary

Let $G$ be a finite group and $k$ a field with $char k ∤ | G |$ which is a splitting field for $G$ . Then $k G$ is a finite product of full matrix algebras over $k$ , and its centre $Z (k G)$ is an $r$ -dimensional commutative $k$ -algebra with two natural bases: the primitive central idempotents $e_{1}, \dots, e_{r}$ attached to the simple modules, and the class sums $C_{g}$ attached to the conjugacy classes.

Lam's $(8.15)$ is the change-of-basis matrix between them, and both directions are written purely in terms of the irreducible characters $χ_{1}, \dots, χ_{r}$ . Every later arithmetic result in this stream — the orthogonality relations, the divisibility $n_{i} ∣ | G |$ , the classification of central torsion units of $A G$ — is obtained by squeezing these two formulas.

2Bases of

Z (k G)

$r$ Blocks

=

classes

$n_{i} / | G |$ Scale factor in

e_{i}

$n_{i} \neq 0$ Forced in

k

Overview

The Wedderburn decomposition of a semisimple ring is governed by its primitive central idempotents: writing $1 = e_{1} + \dots + e_{r}$ as a sum of orthogonal primitive central idempotents is exactly the same data as writing the ring as a product of simple rings. The page Central Idempotents and Ring Direct Decompositions treats that correspondence in general. For a group algebra there is a competing, entirely combinatorial description of the same centre.

For any commutative ring $k$ and any finite group $G$ , an element of $k G$ is central precisely when its coefficient function is constant on conjugacy classes. Hence $Z (k G)$ is free on the class sums $C_{g}$ . When $k G$ is split semisimple, $Z (k G)$ is also $k^{r}$ with the $e_{i}$ as the standard idempotent basis. The number of conjugacy classes and the number of simple modules therefore agree — the fact recorded in the Structure of kG modulo Its Radical page.

Z (k G) = ⨁_{g} k C_{g} = ⨁_{i = 1}^{r} k e_{i},

(8.14)

Two bases of the same $r$ -dimensional commutative algebra; the sum on the left runs over a set of representatives of the conjugacy classes.

The bridge in both directions is the regular character. The class-sum basis is visible to the group; the idempotent basis is visible to the ring; the character table is the dictionary.

Learning Objectives

State the standing hypotheses: $G$ finite, $char k ∤ | G |$ , $k$ a splitting field for $G$ .
Identify the two bases ${e_{i}}$ and ${C_{g}}$ of $Z (k G)$ and explain why both have size $r$ .
Prove $e_{i} = | G |^{- 1} n_{i} \sum_{g} χ_{i} (g^{- 1}) g$ using the regular character.
Prove $C_{g} = m_{g} \sum_{i} χ_{i} (g) n_{i}^{- 1} e_{i}$ by applying $χ_{j}$ to both sides.
Deduce that $char k$ divides no character degree $n_{i}$ .
Compute the three central idempotents of $ℚ S_{3}$ and check them against the class sums.

Definitions

Standing hypotheses for this page: G is a finite group, k is a field whose characteristic does not divide the order of G, and k is a splitting field for G. All modules are finite-dimensional over k.

Definition(8.14)Class sum

For $g \in G$ let $C_{g} = \sum_{x \in 𝒦 (g)} x \in k G$ , where $𝒦 (g)$ is the conjugacy class of $g$ and $m_{g} = | 𝒦 (g) |$ . An element $α = \sum_{g} α_{g} g$ lies in $Z (k G)$ if and only if the function $g \mapsto α_{g}$ is constant on conjugacy classes; hence the distinct class sums form a $k$ -basis of $Z (k G)$ . This holds over any commutative coefficient ring $k$ and needs no hypothesis on the characteristic.

$M_{1}, \dots, M_{r}$: A complete set of pairwise non-isomorphic simple left $k G$ -modules.
$n_{i}$: $\dim_{k} M_{i}$ . Because $k$ splits $G$ , ${End}_{k G} (M_{i}) = k$ and $n_{i}$ is also the matrix size in the Wedderburn decomposition.
$χ_{i}$: The character of $M_{i}$ , extended $k$ -linearly from $G$ to $k G$ ; $χ_{i} (1) = n_{i}$ .
$e_{i}$: The primitive central idempotent of $k G$ acting as the identity on $M_{i}$ and as $0$ on $M_{j}$ for $j \neq i$ .
$χ_{reg}$: The character of $_{k G} k G$ . Explicitly $χ_{reg} (1) = | G |$ and $χ_{reg} (g) = 0$ for $g \neq 1$ .
$ω_{i}$: The central character $Z (k G) \to k$ , the $i$ -th coordinate projection along $Z (k G) ≅ k^{r}$ .

k G ≅ M_{n_{1}} (k) \times \dots \times M_{n_{r}} (k), \sum_{i = 1}^{r} n_{i}^{2} = | G | .

(8.1)

The split semisimple case of the structure theorem; $e_{i}$ is the identity of the $i$ -th factor.

Core Concepts

The regular character as a measuring device

Since $char k ∤ | G |$ , Maschke's Theorem makes $k G$ semisimple, and the splitting hypothesis gives $_{k G} k G ≅ n_{1} M_{1} \oplus \dots \oplus n_{r} M_{r}$ . Taking characters,

χ_{reg} = \sum_{i = 1}^{r} n_{i} χ_{i}, χ_{reg} (g) = {\begin{cases} | G | & g = 1, \\ 0 & g \neq 1 . \end{cases}

(8.15a)

The second description holds because left multiplication by $g \neq 1$ permutes the basis $G$ with no fixed point, so its matrix has zero diagonal.

The two descriptions of $χ_{reg}$ pull in opposite directions: one is module-theoretic, the other reads off a single coefficient. Equating them is a coefficient-extraction device — apply $χ_{reg} (- g^{- 1})$ to any $α \in k G$ and $| G | α_{g}$ falls out. That single trick proves $(8.15) (1)$ .

Idempotents act as Kronecker deltas

By construction $e_{i}$ acts on $M_{j}$ as $δ_{i j}$ times the identity. Applying characters, $χ_{j} (e_{i}) = δ_{i j} n_{j}$ , and more generally $χ_{j} (e_{i} α) = δ_{i j} χ_{j} (α)$ for every $α \in k G$ . This is the only property of the $e_{i}$ that the proofs use.

Simple module

M_{i}

Idempotent

e_{i}

Character

χ_{i}

Coefficients of

e_{i}

Central characters

Restricting the projection $k G \to M_{n_{i}} (k)$ to the centre gives a $k$ -algebra homomorphism $ω_{i} : Z (k G) \to k$ with $ω_{i} (e_{j}) = δ_{i j}$ . Its value on a class sum is forced: $C_{g}$ acts on $M_{i}$ as a scalar, and taking traces of that scalar action gives

ω_{i} (C_{g}) = \frac{χ_{i} (C_{g})}{n_{i}} = \frac{m_{g} χ_{i} (g)}{n_{i}} .

(8.15b)

The quantity whose integrality drives the whole of the next two pages.

Note that $n_{i}$ appears in a denominator, so the formula is only meaningful once we know $n_{i} \neq 0$ in $k$ — which is itself a corollary of $(8.15) (1)$ , not an assumption.

Key Results

Proposition(8.15)The two change-of-basis formulas

Let $G$ be a finite group and let $k$ be a splitting field for $G$ with $char k ∤ | G |$ . With the notation above:

$e_{i} = \frac{n_{i}}{| G |} \sum_{g \in G} χ_{i} (g^{- 1}) g$ for $1 \leq i \leq r$ . Since conjugate elements have equal character values, the right-hand side is a $k$ -combination of class sums. In particular $char k$ does not divide $n_{i}$ for any $i$ .
$C_{g} = m_{g} displaystyle \sum_{i = 1}^{r} \frac{χ_{i} (g)}{n_{i}} e_{i}$ for every $g \in G$ , where $m_{g}$ is the size of the conjugacy class of $g$ .

Proof

(1). Write $e_{i} = \sum_{h \in G} a_{i h} h$ with $a_{i h} \in k$ ; this is possible since $G$ is a $k$ -basis of $k G$ . Fix $g \in G$ and evaluate $χ_{reg}$ at $e_{i} g^{- 1}$ in two ways.

First, by linearity and $χ_{reg} (h g^{- 1}) = | G | δ_{h g}$ ,

χ_{reg} (e_{i} g^{- 1}) = \sum_{h \in G} a_{i h} χ_{reg} (h g^{- 1}) = a_{i g} | G | .

Second, using $χ_{reg} = \sum_{j} n_{j} χ_{j}$ together with $χ_{j} (e_{i} α) = δ_{i j} χ_{j} (α)$ ,

χ_{reg} (e_{i} g^{- 1}) = \sum_{j = 1}^{r} n_{j} χ_{j} (e_{i} g^{- 1}) = n_{i} χ_{i} (g^{- 1}) .

Comparing gives $a_{i g} = | G |^{- 1} n_{i} χ_{i} (g^{- 1})$ , which is (1). If $char k$ divided $n_{i}$ then every coefficient $a_{i g}$ would vanish and $e_{i} = 0$ , contradicting the fact that $e_{i}$ acts as the identity on the nonzero module $M_{i}$ .

(2). $C_{g}$ is central, so $C_{g} = \sum_{i} b_{g, i} e_{i}$ for unique $b_{g, i} \in k$ . Apply $χ_{j}$ . On the left, $χ_{j}$ is constant on the conjugacy class of $g$ , so $χ_{j} (C_{g}) = m_{g} χ_{j} (g)$ . On the right, $χ_{j} (e_{i}) = δ_{i j} n_{j}$ , so $χ_{j} (\sum_{i} b_{g, i} e_{i}) = b_{g, j} n_{j}$ . Hence $b_{g, j} = m_{g} χ_{j} (g) / n_{j}$ , which is legitimate because $n_{j}$ is invertible in $k$ by (1).

Corollary(8.15c)Coefficient of the identity

Under the same hypotheses, the coefficient of $1$ in $e_{i}$ is $n_{i}^{2} / | G |$ . Summing over $i$ and using $\sum_{i} e_{i} = 1$ recovers $\sum_{i = 1}^{r} n_{i}^{2} = | G |$ .

Proof

Put $g = 1$ in $(8.15) (1)$ : the coefficient is $| G |^{- 1} n_{i} χ_{i} (1) = n_{i}^{2} / | G |$ . The identity $e_{1} + \dots + e_{r} = 1$ compares coefficients of $1$ on both sides. This is the same count as $\dim_{k} k G = \sum_{i} n_{i}^{2}$ from the Wedderburn decomposition, now obtained as a by-product.

Corollary(8.15d)Central characters

Under the same hypotheses, the $k$ -algebra homomorphisms $Z (k G) \to k$ are exactly $ω_{1}, \dots, ω_{r}$ , and $ω_{i} (C_{g}) = m_{g} χ_{i} (g) / n_{i}$ . The matrix $(ω_{i} (C_{g_{j}}))$ is invertible, being the change of basis between ${C_{g}}$ and ${e_{i}}$ .

Remark—Both hypotheses are used

Semisimplicity ( $char k ∤ | G |$ ) is what makes $χ_{reg} = \sum_{i} n_{i} χ_{i}$ available and what makes $| G |$ invertible. The splitting hypothesis is what makes $n_{i} = \dim_{k} M_{i}$ and $χ_{i} (e_{i}) = n_{i}$ ; without it one must work with $D_{i} = {End}_{k G} (M_{i})$ and reduced characters, and the primitive central idempotents of $k G$ are Galois sums of the $e_{i}$ computed over a splitting field.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Extract a coefficient with $χ_{reg}$

For $α \in k G$ and $g \in G$ , $χ_{reg} (α g^{- 1}) = | G | α_{g}$ . This turns a module-theoretic quantity into a single coordinate.

Move 2

Apply a character to a central identity

Both sides of an equation in $Z (k G)$ can be hit with $χ_{j}$ ; idempotents collapse to $δ_{i j} n_{j}$ and class sums collapse to $m_{g} χ_{j} (g)$ . Two lines replace a computation.

Move 3

Read a non-vanishing off a formula

If an explicit formula for a nonzero object carries a scalar factor, that factor is nonzero. This is how $char k ∤ n_{i}$ is obtained at no cost.

The structural point is that $Z (k G)$ is small — dimension $r$ , not $| G |$ — yet it sees the whole representation theory, because a semisimple algebra is determined by the idempotents in its centre. Restricting attention to the centre is what makes these arguments two lines rather than two pages.

Worked Example

Take $G = S_{3}$ and $k = ℚ$ . Then $char k = 0$ and $ℚ$ is a splitting field for $S_{3}$ , since $ℚ S_{3} ≅ ℚ \times ℚ \times M_{2} (ℚ)$ .

Character table of $S_{3}$ ; the first column is the degree.
	$1$	$(1 2)$	$(1 2 3)$
class size $m_{g}$	$1$	$3$	$2$
$χ_{1}$ (trivial)	$1$	$1$	$1$
$χ_{2}$ (sign)	$1$	$- 1$	$1$
$χ_{3}$ (standard)	$2$	$0$	$- 1$

Write $σ = (1 2 3)$ , so the three-cycles are ${σ, σ^{2}}$ , and let $T = (1 2) + (1 3) + (2 3)$ be the class sum of the transpositions. Since every element of $S_{3}$ is conjugate to its inverse, $χ_{i} (g^{- 1}) = χ_{i} (g)$ here, and $(8.15) (1)$ reads:

e_{1} = \frac{1}{6} (1 + T + σ + σ^{2}), e_{2} = \frac{1}{6} (1 - T + σ + σ^{2}),

(E.1)

e_{3} = \frac{2}{6} (2 \cdot 1 + 0 \cdot T - (σ + σ^{2})) = \frac{1}{3} (2 - σ - σ^{2}) .

(E.2)

The coefficient of $1$ is $n_{3}^{2} / | G | = 4 / 6 = 2 / 3$ , as $(8.15 c)$ predicts.

Check the identity $e_{1} + e_{2} + e_{3} = 1$ : the first two sum to $\frac{1}{3} (1 + σ + σ^{2})$ , and adding $\frac{1}{3} (2 - σ - σ^{2})$ gives $\frac{1}{3} \cdot 3 = 1$ .

Check idempotency of $e_{3}$ directly. Put $s = σ + σ^{2}$ ; then $s^{2} = σ^{2} + 2 σ^{3} + σ^{4} = 2 + s$ , so

9 e_{3}^{2} = (2 - s)^{2} = 4 - 4 s + s^{2} = 4 - 4 s + 2 + s = 6 - 3 s = 3 (2 - s) = 9 e_{3} .

Now run $(8.15) (2)$ in the other direction. For the transposition class, $m_{g} = 3$ and the column is $(1, - 1, 0)$ , so $T = 3 (e_{1} / 1 - e_{2} / 1 + 0 \cdot e_{3} / 2) = 3 e_{1} - 3 e_{2}$ ; expanding the right-hand side indeed leaves $T$ . For the three-cycle class, $m_{g} = 2$ and the column is $(1, 1, - 1)$ , so

σ + σ^{2} = 2 (e_{1} + e_{2} - \frac{1}{2} e_{3}) = 2 e_{1} + 2 e_{2} - e_{3},

(E.3)

Expanding: $2 (e_{1} + e_{2}) = \frac{2}{3} (1 + σ + σ^{2})$ and $- e_{3} = - \frac{1}{3} (2 - σ - σ^{2})$ ; the coefficients of $1$ cancel and $σ, σ^{2}$ each acquire coefficient $1$ .

Comparison and Classification

The two bases of $Z (k G)$ compared
	Class sums ${C_{g}}$	Block idempotents ${e_{i}}$
Indexed by	conjugacy classes of $G$	simple $k G$ -modules
Exists over	any commutative ring $k$	$k$ a splitting field, $char k ∤ \| G \|$
Coefficients in	${0, 1}$ — integral, canonical	$\| G \|^{- 1} ℤ [χ_{i}]$ in general
Multiplication	structure constants are class-multiplication numbers in $ℤ_{\geq 0}$	orthogonal: $e_{i} e_{j} = δ_{i j} e_{i}$
Computed from	the group alone	the character table
Sees	the group's conjugacy structure	the ring's simple components

Which parts of $(8.15)$ survive as hypotheses are dropped
	${C_{g}}$ is a basis	${e_{i}}$ is a basis	$(8.15) (1)$ holds	$n_{i}$ invertible
$char k = 0$ , $k$ splits $G$	yes	yes	yes	yes
$char k = p ∤ \| G \|$ , $k$ splits $G$	yes	yes	yes	yes
$char k ∤ \| G \|$ , $k$ does not split $G$	yes	partial	no	partial
$char k = p ∣ \| G \|$	yes	no	no	no
$k$ a commutative ring, $G$ finite	yes	no	no	no

Which parts of $(8.15)$ survive as hypotheses are dropped

In row three the primitive central idempotents of $k G$ still exist — $k G$ is semisimple — but they are sums of the $e_{i}$ computed over a splitting field, grouped into Galois orbits, and the coefficient formula acquires a trace. In row four $rad (k G) \neq 0$ and one works with block idempotents in the modular sense instead.

Relationship Map

The results of this page sit between the structure theory and the arithmetic. Everything to the right of $(8.15)$ below is proved by manipulating its two formulas.

Maschke

(6.1)

Wedderburn

(3.5)

Splitting field

(8.2)

(8.15)

Orthogonality

(8.16)

Integrality

(8.18)

$(8.15) (1)$ : $e_{i}$ from characters — coefficient extraction
- gives
  - $char k ∤ n_{i}$
  - $\sum_{i} n_{i}^{2} = | G |$ (coefficient of $1$ )
  - First Orthogonality Relation, on applying $χ_{j}$
$(8.15) (2)$ : $C_{g}$ from idempotents — central characters
- gives
  - $ω_{i} (C_{g}) = m_{g} χ_{i} (g) / n_{i}$
  - Second Orthogonality Relation, on substituting (1)
  - Integrality of $m_{g} χ_{i} (g) / n_{i}$ over $ℤ$

$k G$ dimension

| G |

$Z (k G)$ dimension

r

$\sum_{i} ℤ C_{g}$ the integral centre

Z (ℤ G)

— a ring, finitely generated over

ℤ

${0, e_{i}, \dots}$ the finitely many idempotents of

Z (k G)

Standards and Notation

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

Lam's notation

e_{i}

χ_{i}

C_{g}

m_{g}

n_{i}

r

Common variants

e (χ)

e_{χ}

for the block idempotent;

Irr (G)

for the set of irreducible characters

Class sums

C_{g}

K_{g}

\hat{K}

𝒞^{+}

depending on the source

Central character

ω_{i}

, sometimes

ω_{χ}

λ_{χ}

Degrees

n_{i} = χ_{i} (1)

; written

\deg χ_{i}

χ_{i} (1)

in character-theoretic texts

GAPIrr(G), ConjugacyClasses(G), PrimitiveCentralIdempotentsByCharacterTable

Magma / SageCharacterTable(G), G.character_table()

MarkupPresentation MathML per ISO/IEC 40314; symbol conventions per ISO 80000-2

Computational Notes

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

$(8.15) (1)$ is an algorithm: given the character table of $G$ , each $e_{i}$ costs $O (| G |)$ field operations to write down, and only $r$ distinct coefficients need computing because the coefficient function is a class function.

Getting the table. The Burnside–Dixon–Schneider algorithm computes the ordinary character table without constructing a single representation: it diagonalises the class-multiplication matrices acting on $Z (ℂ G)$ , working modulo a suitable prime $p$ with $p \equiv 1 (\mod \exp G)$ and lifting. The eigenvectors are exactly the rows $(ω_{i} (C_{g_{j}}))$ of $(8.15 d)$ .
Cost. The dominant costs are $r \times r$ eigenvector computations and the class-multiplication coefficients; both are polynomial in $r$ and $| G |$ once the classes are known. Computing conjugacy classes of a permutation or matrix group is the practical bottleneck.
Rational blocks. When $k = ℚ$ the primitive central idempotents are the Galois sums $\sum_{σ} e_{σ (i)}$ ; GAP's wedderga package computes them, and the Wedderburn components, without going through $ℂ$ .
Verification. Cheap checks: $e_{i}^{2} = e_{i}$ , $e_{i} e_{j} = 0$ , $\sum_{i} e_{i} = 1$ , and coefficient of $1$ in $e_{i}$ equal to $n_{i}^{2} / | G |$ . Any transcription error in a character table is caught by one of these.

Failure Modes and Common Mistakes

Do not drop the inverse: $\sum_{g} χ_{i} (g) g$ is not $| G | n_{i}^{- 1} e_{i}$ in general; it is the image of $e_{i}$ under the antipode $g \mapsto g^{- 1}$ .
Do not read $χ_{i} (g) / n_{i}$ as a character value — it is a central character value, an eigenvalue of the scalar action of $C_{g}$ divided by $m_{g}$ .
Do not assume $e_{i} \in ℤ G$ . It almost never is; the integral analogue of this decomposition is the subject of the Integral Group Rings page and behaves quite differently.
Do not confuse the $r$ primitive central idempotents with the $\sum_{i} n_{i}$ primitive idempotents of $k G$ — the latter are not central once some $n_{i} > 1$ , and are far from unique.
When $G$ is abelian all $n_{i} = 1$ and $(8.15) (1)$ becomes the classical Fourier idempotent $| G |^{- 1} \sum_{g} χ (g^{- 1}) g$ ; do not carry the $n_{i}$ over from the abelian case by habit.

Quick Reference

Hypotheses

G

finite,

char k ∤ | G |

k

splits

G

Idempotent

e_{i} = \frac{n_{i}}{| G |} \sum_{g \in G} χ_{i} (g^{- 1}) g

Class sum

C_{g} = m_{g} \sum_{i} \frac{χ_{i} (g)}{n_{i}} e_{i}

Central character

ω_{i} (C_{g}) = \frac{m_{g} χ_{i} (g)}{n_{i}}

Regular character

χ_{reg} = \sum_{i} n_{i} χ_{i}

;

χ_{reg} (g) = | G | δ_{g, 1}

Degrees

char k ∤ n_{i}

and

\sum_{i} n_{i}^{2} = | G |

Coefficient of $1$

n_{i}^{2} / | G |

e_{i}

Orthogonality

e_{i} e_{j} = δ_{i j} e_{i}

\sum_{i} e_{i} = 1

Which formula to reach for
You know	You want	Use
Character table	the block idempotents	$(8.15) (1)$
Block idempotents	the class sums	$(8.15) (2)$
A central element $α$	its coefficient at $g$	$α_{g} = \| G \|^{- 1} χ_{reg} (α g^{- 1})$
A central element $α$	its scalar action on $M_{i}$	$ω_{i} (α) = χ_{i} (α) / n_{i}$
Class sizes and degrees	an arithmetic constraint	$m_{g} χ_{i} (g) / n_{i}$ is an algebraic integer $(8.18)$

Frequently Asked Questions

Why must the ground field be a splitting field, when the class sums are a basis of the centre over any field?

Because the two bases must have the same size. Over any field with $char k ∤ | G |$ the centre has dimension $r$ , the number of conjugacy classes, but the number of simple $k G$ -modules is only $r$ when $k$ splits $G$ ; in general it is $\sum_{i} \dim_{k} Z (D_{i}) = r$ only in the split case. For $ℚ C_{3}$ there are three classes but two simple modules, and the idempotent basis is genuinely coarser.

Does

(8.15) (1)

say the central idempotents are determined by the character table?

Yes, completely — given the character table and a labelling of the conjugacy classes, every primitive central idempotent is written down with no further information about G. This is why character tables are the standard compressed encoding of a group's representation theory, and why computing them is a primitive operation in computational group theory.

How is the corollary that the characteristic does not divide any degree consistent with the modular theory?

It is not a statement about arbitrary $k G$ ; it uses $char k ∤ | G |$ . In that situation every $n_{i}$ divides $| G |$ in characteristic zero and the degrees carry over, so a prime not dividing $| G |$ cannot divide any $n_{i}$ . When $char k$ does divide $| G |$ the simple modules are different objects and their dimensions can be divisible by the characteristic.

What replaces these formulas over the integers?

Nothing as clean. The $e_{i}$ have denominators dividing $| G |$ , so $ℤ G$ typically has no idempotents at all besides $0$ and $1$ — that is exactly Theorem $(8.26)$ on the Integral Group Rings page. The integral object that survives is $Z (ℤ G) = ⨁_{g} ℤ C_{g}$ , a commutative ring finitely generated over $ℤ$ , and its integrality is what powers the divisibility theorems.

Is the class-sum basis multiplicative in a usable way?

Yes: $C_{g} C_{h} = \sum_{x} a_{g h x} C_{x}$ with $a_{g h x}$ non-negative integers, the class multiplication coefficients. Applying $ω_{i}$ turns this into a numerical identity among the quantities $m_{g} χ_{i} (g) / n_{i}$ , which is precisely how the Burnside–Dixon algorithm recovers a character table from the class structure alone.

Where does the factor

n_{i}

come from intuitively?

From the multiplicity of $M_{i}$ in the regular module. The projection of $k G$ onto its $i$ -th Wedderburn component has rank $n_{i}^{2}$ , and the coefficient of $1$ in $e_{i}$ is that rank divided by $| G |$ . The factor $n_{i}$ in the formula is the multiplicity, and the second factor $n_{i}$ hiding in $χ_{i} (1)$ is the dimension.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §8, especially (8.14)–(8.15).
C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, 1962, §§26–33.
I. M. Isaacs, Character Theory of Finite Groups, Academic Press, 1976, Chapters 2 and 3.
W. Feit, The Representation Theory of Finite Groups, North-Holland Mathematical Library 25, 1982.
J. D. Dixon, “High speed computation of group characters”, Numerische Mathematik 10 (1967), 446–450.
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977.

AI Suggested Questions

Derive the First Orthogonality Relation directly from the formula for the central idempotents.
What do the primitive central idempotents of the rational group algebra of the quaternion group of order 8 look like?
How does the Burnside-Dixon-Schneider algorithm recover the central characters from the class multiplication coefficients?
For a non-splitting field, how are the primitive central idempotents obtained as Galois sums, and where does the Schur index enter?
What are the block idempotents of a group algebra in characteristic p dividing the group order, and how are they computed?
Show that an element of the group algebra is central exactly when its coefficient function is a class function, over any commutative coefficient ring.
Compare the central idempotent formula with the Fourier inversion formula for a finite abelian group.

Central Idempotents and Characters

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

The regular character as a measuring device

Idempotents act as Kronecker deltas

Central characters

Key Results

Proof Techniques and Method

Extract a coefficient with $χ_{reg}$

Apply a character to a central identity

Read a non-vanishing off a formula

Worked Example

Comparison and Classification

Relationship Map

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The regular character as a measuring device

Idempotents act as Kronecker deltas

Central characters

06Key Results

07Proof Techniques and Method

Extract a coefficient with χreg

Apply a character to a central identity

Read a non-vanishing off a formula

08Worked Example

09Comparison and Classification

10Relationship Map

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

Key Results

Proof Techniques and Method

Extract a coefficient with $χ_{reg}$

Worked Example

Comparison and Classification

Relationship Map

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions