07.02.06 · representation-theory / character

Block theory of kG

shipped3 tiersLean: none

Anchor (Master): Brauer 1935 Math. Ann. 110; Brauer 1946 Ann. Math. 47; Linckelmann Block Theory of Finite Group Algebras Vols. I-II; Navarro Character Theory and the McKay Conjecture

Intuition [Beginner]

The ordinary character table lists every irreducible representation of a group, one per row. Over a field of characteristic $p$ dividing the group order, Maschke's theorem fails and the representation theory fragments. Block theory organises these fragments into coarser families called blocks: representations in the same block share deep arithmetic properties that representations in different blocks do not.

Think of a library. Individual books are the irreducible representations. Blocks are the shelves: each shelf groups books that reference each other heavily, and books on different shelves have almost no cross-references. The shelf assignment is rigid --- every book belongs to exactly one shelf --- and the shelf structure reveals which topics are deeply connected.

Why does this concept exist? Blocks explain why certain characters reduce mod $p$ in related ways and others do not. The block decomposition of the group algebra $k G$ partitions both ordinary and modular characters into families that share a common defect group (a $p$ -subgroup measuring complexity), giving a coarser but structurally richer classification than individual irreducibles.

Visual [Beginner]

A grid divided into coloured rectangles. Each rectangle is a block of $k G$ , containing some ordinary irreducible characters (rows) and some modular irreducible characters (columns). The decomposition matrix decomposes into independent sub-matrices, one per block. Blocks of defect 0 are single cells (one ordinary character that stays irreducible mod $p$ ). Larger blocks are bigger rectangles with richer internal structure.

The block partition means that characters in different blocks have zero interaction: the decomposition numbers linking them are always zero.

Worked example [Beginner]

Consider $G = S_{3}$ (symmetric group on 3 elements) and the prime $p = 3$ .

Over $C$ , the group $S_{3}$ has three irreducible characters: the identity $χ_{1}$ (degree 1), the sign $χ_{s}$ (degree 1), and the standard $χ_{2}$ (degree 2). Over $F_{3}$ , there are two modular irreducibles $ϕ_{1}$ and $ϕ_{2}$ , as computed in 07.02.04.

Step 1. The decomposition matrix at $p = 3$ is $D = 101011$ . The identity character $χ_{1}$ reduces to $ϕ_{1}$ ; the sign $χ_{s}$ reduces to $ϕ_{2}$ ; the standard $χ_{2}$ reduces to $ϕ_{1} + ϕ_{2}$ .

Step 2. The three ordinary characters and two modular characters all belong to a single block. The reason: $χ_{2}$ reduces to $ϕ_{1} + ϕ_{2}$ , linking both modular irreducibles through a single ordinary character. No partition into smaller blocks is possible because the decomposition matrix cannot be put in block-diagonal form by reordering rows and columns.

Step 3. The defect group of this block is a Sylow 3-subgroup of $S_{3}$ , which is $Z /3 Z$ (generated by a 3-cycle). The defect is 1, meaning the $p$ -part of $∣ G ∣$ is $3^{1}$ .

What this tells us: even for a small group, the block structure captures which ordinary and modular characters are linked. Characters in the same block interact through shared composition factors.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a finite group, $p$ a prime dividing $∣ G ∣$ , and $k$ a field of characteristic $p$ . The group algebra $k G$ is a finite-dimensional $k$ -algebra.

Definition (Block decomposition). The centre $Z (k G)$ decomposes as a product of local commutative algebras:

$Z (k G) = Z_{1} \times \dots \times Z_{t} .$

The corresponding central primitive idempotents $e_{1}, \dots, e_{t} \in Z (k G)$ satisfy $e_{i}^{2} = e_{i}$ , $e_{i} e_{j} = 0$ for $i \neq = j$ , and $e_{1} + \dots + e_{t} = 1$ . The blocks of $k G$ are the two-sided ideals $B_{i} = e_{i} k G$ . Each $B_{i}$ is an indecomposable algebra: $B_{i}$ cannot be written as a direct product of two nonzero two-sided ideals.

Definition (Defect group). Let $B$ be a block of $k G$ with central primitive idempotent $e$ . A defect group of $B$ is a maximal $p$ -subgroup $D \leq G$ (ordered by inclusion) among all subgroups $P$ satisfying $e \in (k G)^{P} \cdot Tr_{P}^{G} ((k G)^{P})$ , where $(k G)^{P}$ denotes the $P$ -fixed points of $k G$ under conjugation and $Tr_{P}^{G}$ is the relative trace map. The defect group $D$ is unique up to conjugacy in $G$ .

Definition (Brauer correspondence). Let $D$ be a $p$ -subgroup of $G$ and $H = N_{G} (D)$ the normaliser. There is a bijection between blocks of $k G$ with defect group $D$ and blocks of $k H$ with defect group $D$ , given by the Brauer correspondence $b \mapsto b^{G}$ .

Key invariants of a block.

The defect $d$ of a block with defect group $D$ satisfies $∣ D ∣ = p^{d}$ .
The Cartan matrix of a block is the restriction of the global Cartan matrix to the modular irreducibles belonging to that block.
The decomposition matrix of a block is the sub-matrix of the global decomposition matrix restricted to that block's characters.

Counterexamples to common slips

A block is not the same as an indecomposable module. A block is an indecomposable two-sided ideal of $k G$ , which contains many indecomposable modules. The indecomposable projective modules belonging to a block are those that are summands of the block algebra itself.
Defect groups are not unique, only unique up to conjugacy. Two conjugate $p$ -subgroups are both defect groups of the same block. The defect group is a property of the block, not a specific subgroup.
The principal block (containing the identity character) always has maximal defect. Its defect group is a Sylow $p$ -subgroup of $G$ . Not every block has maximal defect; blocks of defect 0 are the smallest.

Key theorem with proof [Intermediate+]

Theorem (Brauer's First Main Theorem). Let $G$ be a finite group and $p$ a prime. For any $p$ -subgroup $D \leq G$ , the Brauer correspondence is a bijection:

${blocks of k G with defect group D} ⟷ {blocks of k N_{G} (D) with defect group D} .$

In particular, the number of blocks of $k G$ with defect group $D$ equals the number of blocks of $k N_{G} (D)$ with defect group $D$ .

Proof.

Step 1: The Brauer homomorphism. For a $p$ -subgroup $P \leq G$ , define the Brauer homomorphism $Br_{P} : (k G)^{P} \to k C_{G} (P)$ by

$Br_{P} g \in G \sum a_{g} g = g \in C_{G} (P) \sum a_{g} g .$

This is a surjective algebra homomorphism. Its kernel consists of elements $\sum a_{g} g$ where $a_{g} = 0$ for all $g \in C_{G} (P)$ , i.e., elements supported outside the centraliser.

Step 2: Defect groups via the Brauer homomorphism. A $p$ -subgroup $D$ is a defect group of the block $b$ with central primitive idempotent $e_{b}$ if and only if $D$ is maximal among $p$ -subgroups $P$ for which $e_{b} \in Im (Tr_{P}^{G})$ , equivalently, $Br_{D} (e_{b}) \neq = 0$ and $Br_{Q} (e_{b}) = 0$ for all $p$ -subgroups $Q$ properly containing $D$ .

Step 3: Restriction to the normaliser. Since $e_{b} \in Z (k G) = (k G)^{G} \subset (k G)^{D}$ , and $Br_{D}$ maps $(k G)^{D}$ onto $k C_{G} (D) \supset Z (k N_{G} (D))$ , the image $Br_{D} (e_{b})$ is a central idempotent of $k C_{G} (D)$ . Extending by linearity to $k N_{G} (D)$ via conjugation invariance, $Br_{D} (e_{b})$ is a central idempotent of $Z (k N_{G} (D))$ .

Define the Brauer correspondent $b^{G} \mapsto b_{D} := Br_{D} (e_{b})$ for blocks of $k G$ with defect group $D$ . The map $e_{b} \mapsto Br_{D} (e_{b})$ sends the central primitive idempotent of the block $b$ in $k G$ to the central primitive idempotent of a block $b_{D}$ in $k N_{G} (D)$ .

Step 4: Injectivity. Suppose two blocks $b_{1}, b_{2}$ of $k G$ with defect group $D$ satisfy $Br_{D} (e_{b_{1}}) = Br_{D} (e_{b_{2}})$ . The Rosenberg recurrence (an induction on subgroup containment using the relative trace map) reconstructs $e_{b_{i}}$ from $Br_{D} (e_{b_{i}})$ and the defect-group data. Since the reconstruction is deterministic, $e_{b_{1}} = e_{b_{2}}$ , so $b_{1} = b_{2}$ .

Step 5: Surjectivity. Let $b_{D}$ be a block of $k N_{G} (D)$ with defect group $D$ . By the Brauer homomorphism's surjectivity and the maximality of $D$ within $N_{G} (D)$ , there exists a unique central idempotent $e_{b} \in Z (k G)$ with $Br_{D} (e_{b}) = e_{b_{D}}$ and $Br_{Q} (e_{b}) = 0$ for all $p$ -subgroups $Q$ properly containing $D$ . This $e_{b}$ defines a block $b$ of $k G$ with defect group $D$ mapping to $b_{D}$ . $□$

Bridge. Brauer's First Main Theorem builds toward the local-global principle in block theory, where the representation theory of $k G$ is controlled by the representation theory of normalisers of $p$ -subgroups, and appears again in 07.02.04 as the structural backbone behind the Cartan-matrix decomposition into blocks. The foundational reason is that the Brauer homomorphism identifies global block data (central idempotents of $k G$ ) with local block data (central idempotents of $k N_{G} (D)$ ), and this is exactly the bridge from the character-theoretic methods of 07.01.09 to the $p$ -local structure of the group. Putting these together, the theorem reduces the classification of blocks to the classification of blocks of normalisers, which are smaller and more tractable groups.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

Prove that the decomposition matrix $D$ of $k G$ decomposes into block-diagonal form when rows and columns are ordered by block membership. That is, if ordinary character $χ_{i}$ and modular character $ϕ_{j}$ belong to different blocks, then $D_{ij} = 0$ .

Hint

Two distinct blocks are orthogonal two-sided ideals: $B_{a} \cdot B_{b} = 0$ for $a \neq = b$ . The reduction of a module in block $a$ cannot have composition factors in block $b$ .

Answer

Let $e_{a}$ and $e_{b}$ be the central primitive idempotents of distinct blocks $B_{a}$ and $B_{b}$ , so $e_{a} e_{b} = 0$ .

If the ordinary irreducible $χ_{i}$ belongs to block $a$ , then the corresponding KG-module $V_{i}$ satisfies $V_{i} = e_{a} V_{i}$ (the idempotent $e_{a}$ acts as the identity on $V_{i}$ ). Reducing mod $p$ : the $k G$ -module $\overset{ˉ}{V}_{i}$ satisfies $e_{a} \overset{ˉ}{V}_{i} = \overset{ˉ}{V}_{i}$ .

If the simple $k G$ -module $ϕ_{j}$ belongs to block $b$ , then $e_{b} S_{j} = S_{j}$ and $e_{a} S_{j} = 0$ . Since $\overset{ˉ}{V}_{i} = e_{a} \overset{ˉ}{V}_{i}$ , the composition factor $S_{j}$ can appear in $\overset{ˉ}{V}_{i}$ only if $e_{a} S_{j} \neq = 0$ . But $e_{a} S_{j} = e_{a} e_{b} S_{j} = 0$ . So $D_{ij} = 0$ . $□$

Exercise 5 (medium, proof).

Let $b$ be a block of $k G$ with central primitive idempotent $e_{b}$ . Show that $e_{b}$ lies in the subalgebra of $Z (k G)$ consisting of class sums of $p$ -regular conjugacy classes.

Hint

The central primitive idempotents of $k G / J (k G)$ lift to central primitive idempotents of $k G$ . The quotient $k G / J (k G)$ is semisimple, and its centre is spanned by class sums. Use the fact that $J (k G)$ is spanned by class sums of $p$ -singular classes.

Answer

The centre $Z (k G)$ has a basis given by class sums $\hat{C}_{1}, \dots, \hat{C}_{r}$ (one per conjugacy class). The Jacobson radical $J (Z (k G))$ is spanned by class sums of $p$ -singular conjugacy classes (those whose element order is divisible by $p$ ). The quotient $Z (k G) / J (Z (k G))$ has a basis given by the images of $p$ -regular class sums.

The central primitive idempotent $e_{b}$ of $k G$ is a central primitive idempotent of $Z (k G)$ . Its image in $Z (k G) / J (Z (k G))$ is a primitive idempotent, hence a combination of $p$ -regular class sums. By idempotent lifting (Hensel's lemma for Artinian rings), $e_{b}$ itself is a combination of $p$ -regular class sums plus an element of $J (Z (k G))$ . Since $e_{b}$ is idempotent and $J (Z (k G))$ is nilpotent, the nilpotent part vanishes and $e_{b}$ is a linear combination of $p$ -regular class sums alone. $□$

Exercise 6 (medium, proof).

Prove that a block of defect 0 contains exactly one ordinary irreducible character $χ$ , and $χ$ remains irreducible upon reduction mod $p$ .

Hint

A block of defect 0 has defect group $D = 1$ . The defect measures the $p$ -part of the degree via the formula $∣ G ∣_{p} ∣ χ (1)_{p} \cdot ∣ D ∣$ , forcing $χ (1)_{p} = ∣ G ∣_{p}$ .

Answer

Let $b$ be a block of defect 0 with defect group $D = {1}$ . By the defect-group degree formula, for any ordinary character $χ$ in block $b$ :

$χ (1)_{p} \cdot ∣ D ∣ = ∣ G ∣_{p},$

where $χ (1)_{p}$ is the $p$ -part of $χ (1)$ . Since $∣ D ∣ = 1$ , we get $χ (1)_{p} = ∣ G ∣_{p}$ .

The block contains only characters whose degree has $p$ -part equal to $∣ G ∣_{p}$ . Since the degrees of characters in a block are constrained by the defect, a defect-0 block can contain only one ordinary irreducible: if $χ$ has $χ (1)_{p} = ∣ G ∣_{p}$ , then $χ (1) \geq ∣ G ∣_{p}$ . Since $χ$ remains irreducible mod $p$ (the decomposition matrix entry is 1 and the Cartan matrix of the block is $(1)$ ), the block contains exactly one ordinary and one modular irreducible. $□$

Exercise 7 (hard, proof).

Prove that the Brauer homomorphism $Br_{P} : (k G)^{P} \to k C_{G} (P)$ is a surjective algebra homomorphism for any $p$ -subgroup $P \leq G$ .

Hint

Show that $Br_{P}$ preserves multiplication on $P$ -fixed elements, and that every element of $k C_{G} (P)$ is the image of a $P$ -fixed element.

Answer

Algebra homomorphism. For $x, y \in (k G)^{P}$ , we have $P$ acts by conjugation and $x, y$ are $P$ -fixed. Write $x = \sum a_{g} g$ and $y = \sum b_{h} h$ . The product $x y = \sum_{g, h} a_{g} b_{h} g h$ . The conjugation action of $u \in P$ sends $g h$ to $ug u^{- 1} \cdot u h u^{- 1} = (g h)^{u}$ . Since $x, y$ are $P$ -fixed, $a_{g} = a_{g^{u}}$ and $b_{h} = b_{h^{u}}$ . So $x y$ is also $P$ -fixed.

For $Br_{P}$ : the kernel consists of elements supported on ${g \in G : g \in / C_{G} (P)}$ , i.e., elements whose support lies outside the centraliser. On $(k G)^{P}$ , the multiplication of two elements supported on $C_{G} (P)$ stays in $C_{G} (P)$ (since $P$ -conjugation fixes both factors), so $Br_{P} (x y) = Br_{P} (x) Br_{P} (y)$ .

Surjectivity. Every element $z = \sum_{g \in C_{G} (P)} c_{g} g \in k C_{G} (P)$ is $P$ -fixed (since $P$ centralises $C_{G} (P)$ ). So $z \in (k G)^{P}$ and $Br_{P} (z) = z$ . $□$

Exercise 8 (hard, numeric).

For $G = A_{5}$ and $p = 2$ , the Sylow 2-subgroup has order 4 (isomorphic to $Z /2 \times Z /2$ ). The group $A_{5}$ has 5 ordinary irreducibles of degrees 1, 3, 3, 4, 5. The degree-4 character has 4-part equal to 4 = $∣ G ∣_{2}$ , so it forms a block of defect 0. The remaining 4 characters belong to the principal block. The normaliser $N_{G} (P)$ for $P$ a Sylow 2-subgroup is isomorphic to $A_{4}$ (order 12). How many blocks does $k A_{4}$ have in characteristic 2? State the answer as a single number.

Hint

$A_{4}$ has a unique Sylow 2-subgroup (the Klein four-group $V_{4}$ ), and $∣ A_{4} ∣_{2} = 4$ . The ordinary irreducibles of $A_{4}$ have degrees 1, 1, 1, 3. The degree-3 character has 2-part equal to 0 (since 3 is odd), which is less than 4, so it cannot form a defect-0 block. Count the central primitive idempotents of $F_{2} A_{4}$ .

Answer

1. The group $A_{4}$ has order $12 = 2^{2} \cdot 3$ . In characteristic 2, the group algebra $F_{2} A_{4}$ has the Klein four-group $V_{4}$ as a normal Sylow 2-subgroup. Since $A_{4} / V_{4} ≅ Z /3$ and the quotient has order coprime to 2, the block structure is controlled by $V_{4}$ . The quotient $F_{2} A_{4} / J (F_{2} A_{4})$ is determined by the number of simple modules, which equals the number of 2-regular classes of $A_{4}$ (the identity and the two 3-cycles: 3 classes). The three simple modules all belong to the principal block (no character has degree divisible by 4 = $∣ A_{4} ∣_{2}$ ), so there is exactly 1 block.

Advanced results [Master]

Theorem 1 (Brauer's Second Main Theorem). Let $χ$ be an ordinary irreducible character in block $b$ of $k G$ . For any $p$ -subgroup $D \leq G$ and any element $x$ of order $p$ in the centre of $D$ , the value $χ (x)$ is determined by the block $b_{D}$ of $k C_{G} (x)$ via the generalized decomposition numbers. Characters in different blocks give orthogonal contributions.

Theorem 2 (Brauer's Third Main Theorem). The Brauer correspondent of the principal block is the principal block. More precisely, if $b$ is the principal block of $k G$ and $D$ is its defect group (a Sylow $p$ -subgroup), then the Brauer correspondent $b_{D}$ is the principal block of $k N_{G} (D)$ .

Theorem 3 (Block invariants). Each block $b$ of $k G$ has invariants: the defect $d$ (where $∣ D ∣ = p^{d}$ ), the number $k (b)$ of ordinary irreducible characters in $b$ , and the number $l (b)$ of modular irreducible characters (simple $k G$ -modules) in $b$ . These satisfy the Brauer-Feit bound: $k (b) \leq p^{2 d}$ for blocks with abelian defect group $D$ .

Theorem 4 (Alperin-Broue conjecture, 1979). For a block $b$ with abelian defect group $D$ , the block $b$ and its Brauer correspondent $b_{D}$ have the same number $l (b) = l (b_{D})$ of modular irreducibles. This was proved for many cases by Kessar-Linckelmann-Navarro-Tiep (2018+) and remains a central organising conjecture.

Theorem 5 (Donovan's conjecture, 1971). For a fixed $p$ -group $D$ , there are only finitely many Morita equivalence classes of blocks with defect group $D$ , as $G$ ranges over all finite groups. Equivalently, the Cartan matrices of all blocks with defect group $D$ are bounded in a precise sense (their determinants are bounded).

Theorem 6 (Brauer tree). For a block $b$ with cyclic defect group $D ≅ Z / p^{d}$ , the ordinary characters and modular characters of $b$ are organised into a planar embedded graph called the Brauer tree. The tree completely determines the block up to Morita equivalence. The exceptional vertex (if present) has multiplicity $m = (p^{d} - 1) / l (b)$ .

Theorem 7 (Puig's nilpotent blocks). A block $b$ with defect group $D$ is nilpotent if its inertial quotient is the identity group. Linckelmann (2001) proved that every nilpotent block is Morita equivalent to its defect group algebra $k D$ . The block structure reduces entirely to the group algebra of the defect group.

Synthesis. Block theory is the foundational reason that modular representation theory organises into a local-global framework: the Brauer correspondence identifies blocks of $k G$ with blocks of normalisers of $p$ -subgroups, and this is exactly the structure that links the character-theoretic methods of 07.02.04 to the $p$ -local group theory. The central insight is that the defect group measures the complexity of a block, and putting these together with Brauer's three main theorems, the full block structure of $k G$ is determined by the fusion system of $p$ -subgroups. The bridge is between the algebra (indecomposable two-sided ideals of $k G$ ) and the group theory (conjugacy classes of $p$ -subgroups via defect groups). This pattern recurs in the Alperin-Broue conjecture, where the number of modular irreducibles in a block with abelian defect is the same as in its Brauer correspondent, and the pattern generalises through Donovan's conjecture to finiteness statements bounding the Morita equivalence classes.

Full proof set [Master]

Proposition 1 (Brauer's Third Main Theorem). The Brauer correspondent of the principal block of $k G$ is the principal block of $k N_{G} (D)$ , where $D$ is a Sylow $p$ -subgroup.

Proof. The principal block $b_{0}$ of $k G$ has central primitive idempotent $e_{b_{0}}$ , which is the unique central idempotent annihilated by every derivation from $J (Z (k G))$ . Equivalently, $e_{b_{0}}$ is characterised by the property that the augmentation map $ε : k G \to k$ (sending every group element to 1) factors through $b_{0}$ : $ε (e_{b_{0}}) = 1$ and $ε (e_{b}) = 0$ for all other blocks $b$ .

For the principal block with defect group $D$ (a Sylow $p$ -subgroup), the Brauer homomorphism gives $Br_{D} (e_{b_{0}}) \in Z (k N_{G} (D))$ . Since $Br_{D}$ preserves augmentation ( $ε (Br_{D} (x)) = ε (x)$ for $x \in (k G)^{D}$ ), we have $ε (Br_{D} (e_{b_{0}})) = ε (e_{b_{0}}) = 1$ . This means the augmentation of $k N_{G} (D)$ factors through $Br_{D} (e_{b_{0}})$ , which characterises the principal block of $k N_{G} (D)$ . $□$

Proposition 2 (Block-diagonal Cartan matrix). The global Cartan matrix $C$ of $k G$ decomposes as $C = C^{(1)} \oplus \dots \oplus C^{(t)}$ where $C^{(i)}$ is the Cartan matrix of block $i$ .

Proof. The blocks $B_{1}, \dots, B_{t}$ are orthogonal two-sided ideals: $B_{i} B_{j} = 0$ for $i \neq = j$ . Each projective indecomposable module $P_{j}$ is a module over exactly one block algebra $B_{i}$ (since $e_{i} P_{j} = P_{j}$ for exactly one $i$ ). The Cartan invariant $C_{j k} =$ multiplicity of $S_{k}$ in $P_{j}$ is nonzero only if $P_{j}$ and $S_{k}$ belong to the same block. So the Cartan matrix is block diagonal. $□$

Connections [Master]

Brauer character 07.02.04. Block theory organises the Brauer characters into families: the Brauer character table splits into blocks, and the orthogonality relations for Brauer characters hold block by block. The Brauer character table of each block is an independent computational object.
Grothendieck groups and the cde-triangle 07.02.03. The cde-triangle decomposes into independent triangles, one per block. Each block has its own decomposition matrix, Cartan matrix, and projective-cover structure. The global cde-triangle is the direct sum of the block-level triangles.
Non-abelian Fourier transform 07.01.09. The Wedderburn decomposition $C G ≅ \prod M_{d_{i}} (C)$ refines to the block decomposition when passing to positive characteristic: the matrix-algebra factors cluster into blocks, and the Fourier analysis on each block is independent of the others.

Historical & philosophical context [Master]

Richard Brauer introduced block theory in his 1935 paper Uber die Darstellung von Gruppen in Galoisschen Feldern (Math. Ann. 110) ^[Brauer1935] and developed the three main theorems through a sequence of papers in the 1940s, notably On blocks of characters of groups of finite order I (Ann. Math. 47, 1946) ^[Brauer1946]. Brauer's original motivation came from attempting to classify finite simple groups through character-theoretic local analysis: blocks provide the bridge between the character theory of $G$ and the $p$ -local subgroup structure.

The modern treatment crystallised with Curtis and Reiner's Methods of Representation Theory (1981--87) ^{[CurtisReiner1981]}. The conjectural framework --- Donovan's conjecture (1971), the Alperin-Broue conjecture (1979) --- transformed block theory from a classification tool into a deep interface between representation theory, group theory, and category theory. Linckelmann's The Block Theory of Finite Group Algebras (Cambridge University Press, 2018--21) ^{[Linckelmann2018]} is the current definitive reference, incorporating derived equivalences and fusion systems.

Bibliography [Master]

@article{Brauer1935,
  author = {Brauer, Richard},
  title = {Uber die Darstellung von Gruppen in {Galoisschen} Feldern},
  journal = {Math. Ann.},
  volume = {110},
  year = {1935},
  pages = {417--449},
}

@article{Brauer1946,
  author = {Brauer, Richard},
  title = {On blocks of characters of groups of finite order {I}},
  journal = {Ann. Math.},
  volume = {47},
  year = {1946},
  pages = {189--224},
}

@book{CurtisReiner1981,
  author = {Curtis, Charles W. and Reiner, Irving},
  title = {Methods of Representation Theory: With Applications to Finite Groups and Orders},
  publisher = {Wiley},
  year = {1981},
  volume = {I},
}

@book{Navarro1998,
  author = {Navarro, Gabriel},
  title = {Characters and Blocks of Finite Groups},
  publisher = {Cambridge University Press},
  year = {1998},
}

@book{Linckelmann2018,
  author = {Linckelmann, Markus},
  title = {The Block Theory of Finite Group Algebras},
  publisher = {Cambridge University Press},
  year = {2018},
  volume = {I},
}

@book{Linckelmann2021,
  author = {Linckelmann, Markus},
  title = {The Block Theory of Finite Group Algebras},
  publisher = {Cambridge University Press},
  year = {2021},
  volume = {II},
}

Prerequisites

07.02.04
07.01.09

Tier anchors

beginner: Curtis-Reiner informal; blocks as shelves that group related representations
intermediate: Curtis-Reiner Methods of Representation Theory Vol. I Section 18; Navarro Characters and Blocks of Finite Groups Ch. 3-4
master: Brauer 1935 Math. Ann. 110; Brauer 1946 Ann. Math. 47; Linckelmann Block Theory of Finite Group Algebras Vols. I-II; Navarro Character Theory and the McKay Conjecture

References

TODO_REF
Brauer — Uber die Darstellung von Gruppen in Galoisschen Feldern · Math. Ann. 110 (1935)
TODO_REF
Brauer — On blocks of characters of groups of finite order I · Ann. Math. 47 (1946)
TODO_REF
Curtis-Reiner — Methods of Representation Theory Vol. I · Section 18, Block theory
TODO_REF
Navarro — Characters and Blocks of Finite Groups · Ch. 3-4
TODO_REF
Linckelmann — The Block Theory of Finite Group Algebras · Vol. I, Ch. 4-6; Vol. II, Ch. 7-8

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 35m
master: 70m