07.02.03 · representation-theory / character

Grothendieck groups and the cde-triangle

shipped3 tiersLean: none

Anchor (Master): Grothendieck 1957 Tohoku Math. J. 9; Brauer 1956 Math. Z. 63; Serre Linear Representations Part III; Curtis-Reiner Methods of Representation Theory Vol. I

Intuition [Beginner]

The Grothendieck group is a bookkeeping device that turns the collection of representations of a group into an abelian group where you can both add and subtract. Representations can be added by taking direct sums, but there is no natural way to subtract one representation from another within the category of representations. The Grothendieck group introduces "virtual representations" --- formal differences like $[V] - [W]$ that may not correspond to actual representations but behave like them algebraically.

The construction mirrors how the integers $Z$ arise from the natural numbers $N$ . You can add natural numbers but cannot subtract 5 from 3 within $N$ . The solution: form pairs $(a, b)$ and declare $(a, b)$ equivalent to $(c, d)$ whenever $a + d = b + c$ . The pair $(a, b)$ represents "a minus b." The Grothendieck group does the same for representations: the pair $(V, W)$ represents the virtual representation $[V] - [W]$ .

Why does this concept exist? In modular representation theory --- when the characteristic of the field divides the order of the group --- Maschke's theorem fails and representations no longer decompose cleanly into irreducibles. The cde-triangle uses three Grothendieck groups to relate representations in characteristic zero to representations in positive characteristic, providing the structural bridge between ordinary and modular representation theory.

Visual [Beginner]

A triangle connecting three abelian groups. The top vertex is $R_{K} (G)$ , the Grothendieck group of ordinary (characteristic-zero) representations. The bottom-left vertex is $P_{k} (G)$ , the Grothendieck group of projective modules over the field of characteristic $p$ . The bottom-right vertex is $R_{k} (G)$ , the Grothendieck group of all modular representations. Three arrows form the triangle: the decomposition map $d$ goes from $R_{K} (G)$ down to $R_{k} (G)$ , the lift map $e$ goes from $P_{k} (G)$ up to $R_{K} (G)$ , and the Cartan map $c$ goes from $P_{k} (G)$ across to $R_{k} (G)$ .

The commuting triangle encodes Brauer's insight: the way representations reduce from characteristic zero to positive characteristic is controlled by the projective modules.

Worked example [Beginner]

Consider the cyclic group $Z /3 Z = {0, 1, 2}$ with addition mod 3, and let $p = 3$ .

Over the complex numbers $C$ , the group $Z /3 Z$ has three irreducible representations, each 1-dimensional. The identity representation sends every element to 1. The $ω$ -representation sends the generator 1 to $ω = e^{2 π i /3}$ . The $ω^{2}$ -representation sends the generator to $ω^{2}$ .

Step 1. The Grothendieck group $R (Z /3 Z)$ over $C$ is the free abelian group with basis given by these three irreducibles. Every element is a formal combination $a \cdot [identity] + b \cdot [ω] + c \cdot [ω^{2}]$ with $a, b, c$ integers. For instance, $[ω] - [identity]$ is a virtual representation of degree 0 whose character takes the value 0 at the identity and $- 2$ at the generator.

Step 2. Over $F_{3}$ (the field with 3 elements), the characteristic equals $∣ Z /3 Z ∣ = 3$ , so Maschke's theorem fails. There is only one irreducible representation over $F_{3}$ : the representation sending every element to 1.

Step 3. The decomposition map sends each ordinary irreducible to the single modular irreducible. The decomposition matrix is a single column of three 1s: $D = (1, 1, 1)^{T}$ .

Step 4. The Cartan matrix is $C = D^{T} D = (3)$ . The projective cover of the unique modular irreducible has composition length 3 --- it is the regular representation $F_{3} [Z /3 Z]$ itself.

What this tells us: the three distinct ordinary irreducibles all collapse to the same modular irreducible when reduced mod 3. The Cartan invariant 3 measures the information lost in this reduction.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a finite group and $k$ a field. The Grothendieck group $R_{k} (G)$ of the category of finitely generated $k G$ -modules is the abelian group generated by symbols $[M]$ for each isomorphism class of finitely generated $k G$ -modules $M$ , subject to the relation $[M] = [M^{'}] + [M^{''}]$ for every short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ of $k G$ -modules.

Equivalent construction. Let $F$ be the free abelian group on the set of isomorphism classes of finitely generated $k G$ -modules. Let $R$ be the subgroup generated by elements of the form $[M] - [M^{'}] - [M^{''}]$ for every short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ . Then $R_{k} (G) = F / R$ .

Key properties.

When $k$ has characteristic 0 (or characteristic coprime to $∣ G ∣$ ), Maschke's theorem 07.02.01 ensures every module is a direct sum of irreducibles. The short-exact-sequence relation reduces to $[V \oplus W] = [V] + [W]$ , and $R_{k} (G)$ is the free abelian group on isomorphism classes of irreducible representations.
When $char (k) = p$ divides $∣ G ∣$ , Maschke fails. By the Jordan-Holder theorem, every module has a composition series with well-defined composition factors. The class $[M]$ in $R_{k} (G)$ equals the sum of classes of its composition factors, so $R_{k} (G)$ is the free abelian group on isomorphism classes of simple $k G$ -modules.

Definition ( $p$ -modular system). A $p$ -modular system for $G$ is a triple $(K, O, k)$ where:

$O$ is a complete discrete valuation ring of characteristic 0 with maximal ideal $m = (π)$ ,
$K = Frac (O)$ is the fraction field, of characteristic 0, containing sufficiently many roots of unity for $G$ ,
$k = O / m$ is the residue field, of characteristic $p$ .

The three Grothendieck groups of the cde-triangle.

$R_{K} (G)$ : the Grothendieck group of finitely generated $K G$ -modules (ordinary representations).
$R_{k} (G)$ : the Grothendieck group of finitely generated $k G$ -modules (modular representations).
$P_{k} (G)$ : the Grothendieck group of finitely generated projective $k G$ -modules.

Counterexamples to common slips

$R_{k} (G)$ is not the free abelian group on indecomposables when $char (k) ∣ ∣ G ∣$ . The Grothendieck group uses short exact sequences, not just direct sums. When Maschke fails, indecomposable modules need not be simple, and the classes of projective indecomposables in $P_{k} (G)$ map to a proper sublattice of $R_{k} (G)$ via the Cartan map.
The decomposition map $d$ need not be injective. The number $s$ of modular irreducibles is at most the number $r$ of ordinary irreducibles. When $s < r$ , the kernel of $d$ is nonzero and captures the ordinary representations that become decomposable upon reduction mod $p$ .

Key theorem with proof [Intermediate+]

Theorem (Brauer's cde-triangle). Let $(K, O, k)$ be a $p$ -modular system for the finite group $G$ . There exist group homomorphisms

$d : R_{K} (G) \to R_{k} (G), e : P_{k} (G) \to R_{K} (G), c : P_{k} (G) \to R_{k} (G),$

such that $c = d \circ e$ . With respect to natural bases, the Cartan matrix $C$ satisfies $C = D^{T} D$ where $D$ is the decomposition matrix.

Proof.

Step 1: The decomposition map $d$ . Let $M$ be a finitely generated $K G$ -module. Since $O$ is a complete DVR with fraction field $K$ , there exists a finitely generated $O G$ -lattice $L \subset M$ with $K \cdot L = M$ (clear denominators of a $K$ -basis). Define

$d ([M]) := [L / π L] \in R_{k} (G) .$

For well-definedness: let $L^{'}$ be another $O G$ -lattice in $M$ . By the theory of lattices over orders, there exists nonzero $a \in O$ with $a L \subset L^{'} \subset L$ . The exact sequence $0 \to L^{'} / a L \to L / a L \to L / L^{'} \to 0$ has all three terms annihilated by powers of $π$ . Tensoring with $k = O / (π)$ and using the short-exact-sequence relations in $R_{k} (G)$ , the classes $[L / π L]$ and $[L^{'} / π L^{'}]$ coincide. Additivity in short exact sequences extends $d$ to a group homomorphism.

Step 2: The Cartan map $c$ . Every projective $k G$ -module is in particular a $k G$ -module. Define $c : P_{k} (G) \to R_{k} (G)$ by $c ([P]) = [P]$ , forgetting that $P$ is projective. This is a group homomorphism.

Step 3: The lift map $e$ . Let $P$ be an indecomposable projective $k G$ -module. By the theory of projective covers and Hensel's lemma applied to the complete DVR $O$ , there exists a projective $O G$ -module $\hat{L}$ with $\hat{L} / π \hat{L} ≅ P$ . Define

$e ([P]) := [K \otimes_{O} \hat{L}] \in R_{K} (G) .$

Since $P$ is projective, $\hat{L}$ is unique up to isomorphism by Nakayama's lemma, so $e$ is well-defined. By the Krull-Schmidt theorem, $e$ extends to a group homomorphism $P_{k} (G) \to R_{K} (G)$ .

Step 4: Commutativity $c = d \circ e$ . For any $[P] \in P_{k} (G)$ with projective lift $\hat{L}$ :

$d (e ([P])) = d ([K \otimes_{O} \hat{L}]) = [\hat{L} / π \hat{L}] = [P] = c ([P]) .$

The second equality uses the definition of $d$ applied to the lattice $\hat{L}$ inside $K \otimes_{O} \hat{L}$ . The third equality is $\hat{L} / π \hat{L} ≅ P$ by construction.

Step 5: The matrix relation $C = D^{T} D$ . Choose bases ${χ_{1}, \dots, χ_{r}}$ for $R_{K} (G)$ , ${ϕ_{1}, \dots, ϕ_{s}}$ for $R_{k} (G)$ , and ${P_{1}, \dots, P_{s}}$ for $P_{k} (G)$ . The decomposition matrix $D$ is the $r \times s$ matrix with $D_{ij}$ = the multiplicity of $ϕ_{j}$ in $d (χ_{i})$ . Brauer reciprocity (Proposition 1 in the Full proof set below) states that the coefficient of $χ_{i}$ in $e (P_{j})$ equals $D_{ij}$ . Writing elements as row vectors, $d$ acts by right multiplication by $D$ and $e$ by right multiplication by $D^{T}$ . Since $c = d \circ e$ , the matrix of $c$ is $D^{T} \cdot D$ , an $s \times s$ matrix. $□$

Bridge. The commutativity $c = d \circ e$ builds toward 07.02.04, where the decomposition map is computed explicitly on $p$ -regular elements via Brauer characters, and appears again in 07.01.11, which constructs $R_{K} (G)$ from monomial characters of elementary subgroups. The foundational reason is that the cde-triangle identifies the combinatorial data of ordinary characters with the homological data of projective modules, and this is exactly the bridge between ordinary and modular representation theory. Putting these together, the identity $C = D^{T} D$ generalises the orthogonality relations of 07.01.04 to the modular setting.

Exercises [Intermediate+]

Exercise 5 (medium, proof).

Let $G$ be a $p$ -group and $k$ a field of characteristic $p$ . Show that $k G$ has a unique simple module (the module on which every group element acts as the identity) and that the Cartan matrix is the $1 \times 1$ matrix $C = (∣ G ∣)$ .

Hint

For a $p$ -group in characteristic $p$ , the Jacobson radical $J (k G)$ is the augmentation ideal. The unique simple module is $k G / J (k G) ≅ k$ , and the unique projective indecomposable is the regular module $k G$ .

Answer

Let $G$ be a $p$ -group and $k$ a field of characteristic $p$ . The augmentation ideal $I = ker (ε : k G \to k)$ , where $ε (\sum a_{g} g) = \sum a_{g}$ , is a nilpotent ideal (since $G$ is a $p$ -group, $I^{∣ G ∣} = 0$ ). So $I = J (k G)$ is the Jacobson radical.

The unique simple $k G$ -module is $k G / J (k G) ≅ k$ with the action where every $g \in G$ acts as 1. The unique projective indecomposable is $k G$ itself (the regular module). Its composition factors are all copies of $k$ , and the number of copies is $dim_{k} (k G) = ∣ G ∣$ .

The decomposition matrix is the column vector $D = (1, 1, \dots, 1)^{T}$ of length $r$ (the number of ordinary irreducibles, which is $∣ G ∣$ , the number of conjugacy classes). The Cartan matrix is $C = D^{T} D = (r) = (∣ G ∣)$ .

Exercise 7 (hard, proof).

Prove that the Cartan matrix $C$ is invertible over $Z$ (i.e., $det (C) \neq = 0$ ). Conclude that the Cartan map $c : P_{k} (G) \to R_{k} (G)$ is injective.

Hint

Show that the columns of $D$ are linearly independent, then deduce that $D^{T} D$ is invertible.

Answer

The columns of $D$ correspond to the simple $k G$ -modules $ϕ_{1}, \dots, ϕ_{s}$ . Each $ϕ_{j}$ appears as a composition factor of the reduction of at least one ordinary irreducible $χ_{i}$ (otherwise $ϕ_{j}$ would not be a composition factor of any module, contradicting simplicity). So each column of $D$ has at least one nonzero entry.

More strongly, the columns of $D$ are linearly independent. For if $\sum_{j} α_{j} D_{\cdot, j} = 0$ (a linear relation among columns), then for each $i$ : $\sum_{j} α_{j} d_{ij} = 0$ . This says $d (\sum_{j} α_{j} ϕ_{j}) = 0$ in... Actually, the cleaner argument: the images $d (χ_{1}), \dots, d (χ_{r})$ span $R_{k} (G)$ (Exercise 2), and $R_{k} (G)$ is free of rank $s$ . So the $r \times s$ matrix $D$ has rank $s$ , meaning its columns are linearly independent.

Since $D$ has rank $s$ , the $s \times s$ matrix $C = D^{T} D$ has rank $s$ , hence is invertible. The Cartan map $c$ has matrix $C$ , which is invertible, so $c$ is injective (in fact, an isomorphism of free abelian groups of the same rank). $□$

Exercise 8 (hard, proof).

Let $G = S_{3}$ and $p = 2$ . The 2-regular conjugacy classes of $S_{3}$ are ${e}$ and ${(123), (132)}$ , giving $s = 2$ simple $F_{2} S_{3}$ -modules. The ordinary irreducible characters restrict to 2-regular elements as $χ_{1} ∣ = (1, 1)$ , $χ_{s} ∣ = (1, 1)$ , $χ_{2} ∣ = (2, - 1)$ . Compute the decomposition matrix $D$ and verify $C = D^{T} D$ .

Hint

The two simple $F_{2} S_{3}$ -modules are the identity module $ϕ_{1}$ and a 2-dimensional simple $ϕ_{2}$ . Since $χ_{1}$ and $χ_{s}$ agree on 2-regular elements, they reduce to the same modular irreducible. Write $d (χ_{1}) = α ϕ_{1} + β ϕ_{2}$ and solve for $α, β$ using the character values.

Answer

The Brauer characters of the two simple modules on 2-regular elements are $ϕ_{1} = (1, 1)$ and $ϕ_{2} = (2, - 1)$ (the 2-dimensional irreducible; the element $(123)$ has eigenvalues $ω, ω^{2}$ which lift to $ϕ_{2} ((123)) = ω + ω^{2} = - 1$ ).

For $χ_{1}$ (identity, values $(1, 1)$ on 2-regular elements): $d (χ_{1}) = ϕ_{1}$ since both have the same values. So $d_{11} = 1, d_{12} = 0$ .

For $χ_{s}$ (sign, values $(1, 1)$ on 2-regular elements, since $(123)$ is even): $d (χ_{s}) = ϕ_{1}$ . So $d_{21} = 1, d_{22} = 0$ .

For $χ_{2}$ (2-dim, values $(2, - 1)$ ): $χ_{2} ∣ = (2, - 1) = ϕ_{2}$ , so $d (χ_{2}) = ϕ_{2}$ . So $d_{31} = 0, d_{32} = 1$ .

$D = 110001, C = D^{T} D = (101001) 110001 = (2001) .$

The projective cover $P_{1}$ has composition factors $2 ϕ_{1}$ (a non-split extension), and $P_{2} = ϕ_{2}$ is already projective. The determinant $det (C) = 2$ .

Advanced results [Master]

Theorem 1 (Representation ring). The Grothendieck group $R (G) = R_{C} (G)$ carries a ring structure induced by the tensor product of representations: $[V] \cdot [W] = [V \otimes W]$ . With this multiplication, $R (G)$ is a commutative ring with identity $[1]$ (the class of the identity representation). The character map $ch : R (G) \to Class (G, C)$ is an injective ring homomorphism whose image is the subring of virtual characters.

Theorem 2 (Adams operations). For each $k \geq 1$ , the Adams operation $ψ^{k} : R (G) \to R (G)$ is the ring homomorphism defined on characters by $ψ^{k} (χ) (g) = χ (g^{k})$ . These satisfy $ψ^{k} \circ ψ^{l} = ψ^{k l}$ and $ψ^{p} \equiv id$ modulo the $p$ -ideal $I_{p} = ker (R (G) \to R_{k} (G))$ . The Adams operations connect the representation ring to stable homotopy theory via the $J$ -homomorphism.

Theorem 3 (Artin induction in $R (G)$ ). Artin's induction theorem 07.01.10 becomes: $R (G) \otimes_{Z} Q$ is generated by characters induced from cyclic subgroups of $G$ . Equivalently, every element of $R (G)$ is a $Q$ -linear combination of characters induced from 1-dimensional characters of cyclic subgroups.

Theorem 4 (Brauer induction in $R (G)$ ). Brauer's induction theorem 07.01.11 sharpens Artin's result: every element of $R (G)$ is a $Z$ -linear combination of characters induced from elementary subgroups (direct products of a $p$ -group with a cyclic group of order coprime to $p$ ).

Theorem 5 (Blocks and the block decomposition). The group algebra $k G$ decomposes as a direct product of indecomposable two-sided ideals $B_{1} \oplus \dots \oplus B_{t}$ called blocks. Each block $B_{i}$ has an associated Cartan matrix $C^{(i)}$ (the restriction of the global Cartan matrix to that block), and the global Cartan matrix is block diagonal: $C = C^{(1)} \oplus \dots \oplus C^{(t)}$ . Blocks of defect 0 correspond to ordinary irreducibles that remain irreducible upon reduction mod $p$ and have Cartan matrix $(1)$ .

Theorem 6 (Symmetry and definiteness of $C$ ). The identity $C = D^{T} D$ implies $C$ is symmetric and positive semi-definite with $det (C) > 0$ . The determinant $det (C)$ is a power of $p$ (Brauer's defect-group theorem: $p^{d (d - 1) /2}$ for a block of defect $d$ ).

Theorem 7 (The Grothendieck group as $K_{0}$ ). The group $R_{k} (G) = K_{0} (k G)$ is the zeroth K-group of the group algebra. The devissage theorem identifies $K_{0} (k G)$ with $K_{0} (k G / J (k G))$ where $J (k G)$ is the Jacobson radical. The Cartan map $c : P_{k} (G) \to R_{k} (G)$ is the Cartan map of algebraic K-theory.

Theorem 8 (Higher K-theoretic refinement). The cde-triangle lifts to a long exact sequence in algebraic K-theory relating $K_{i} (O G)$ , $K_{i} (K G)$ , and $K_{i} (k G)$ for $i \geq 0$ . At $i = 0$ , this recovers the cde-triangle. At $i = 1$ , it gives information about units and determinants of representations (Swan 1970).

Synthesis. The cde-triangle is the foundational reason that ordinary and modular representation theory communicate: the decomposition map $d$ sends ordinary characters to modular ones, the lift map $e$ recovers ordinary data from projective modules, and the Cartan map $c$ measures the failure of semisimplicity. The central insight is that $C = D^{T} D$ identifies the combinatorial data (decomposition numbers) with the homological data (Cartan invariants). This is exactly the bridge between the character ring $R (G)$ of 07.01.03 and the modular theory of Brauer. Putting these together with Brauer's induction theorem 07.01.11, the representation ring $R (G)$ is generated by monomial characters while its reduction to $R_{k} (G)$ is governed by the Cartan matrix, and the pattern recurs throughout the theory: the Grothendieck group generalises from finite groups to compact Lie groups via the Peter-Weyl theorem 07.07.02, and the bridge is the K-theoretic identification of $R (G)$ with $K_{0}$ of the group algebra.

Full proof set [Master]

Proposition 1 (Brauer reciprocity). The multiplicity of the ordinary irreducible $χ_{i}$ in $e (P_{j})$ equals the decomposition number $d_{ij}$ .

Proof. Let $P_{j}$ be the projective cover of the simple $k G$ -module with character $ϕ_{j}$ , and let $\hat{L}_{j}$ be its projective $O G$ -lift. The $K G$ -module $K \otimes_{O} \hat{L}_{j}$ decomposes:

$K \otimes_{O} \hat{L}_{j} ≅ i ⨁ a_{ij} V_{i}$

for non-negative integers $a_{ij}$ , where $V_{i}$ is the irreducible $K G$ -module with character $χ_{i}$ . The multiplicity $a_{ij}$ equals $dim_{K} Hom_{K G} (V_{i}, K \otimes \hat{L}_{j})$ .

By adjunction between $O G$ -maps and $K G$ -maps (extension of scalars from $O$ to $K$ ):

$dim_{K} Hom_{K G} (V_{i}, K \otimes \hat{L}_{j}) = rank_{O} Hom_{O G} (L_{i}, \hat{L}_{j})$

where $L_{i}$ is any $O G$ -lattice in $V_{i}$ .

Reducing mod $π$ : the $O$ -module $Hom_{O G} (L_{i}, \hat{L}_{j})$ is free of finite rank. Tensoring with $k$ :

$Hom_{O G} (L_{i}, \hat{L}_{j}) \otimes_{O} k ≅ Hom_{k G} (\overset{ˉ}{L}_{i}, P_{j}) .$

Since $P_{j}$ is the projective cover of $ϕ_{j}$ , the dimension of $Hom_{k G} (\overset{ˉ}{L}_{i}, P_{j})$ equals the multiplicity of $ϕ_{j}$ as a composition factor of $\overset{ˉ}{L}_{i} = L_{i} / π L_{i}$ , which is $d_{ij}$ by definition. The isomorphism above preserves dimensions over $k$ , giving $a_{ij} = d_{ij}$ . $□$

Proposition 2 (Block diagonal structure). The group algebra $k G$ decomposes as $k G = B_{1} \oplus \dots \oplus B_{t}$ where each $B_{i}$ is an indecomposable two-sided ideal (a block). The global Cartan matrix is block diagonal with blocks $C^{(1)}, \dots, C^{(t)}$ .

Proof. The center $Z (k G)$ decomposes as $Z (k G) = Z_{1} \oplus \dots \oplus Z_{t}$ where each $Z_{i}$ is a local ring (an indecomposable commutative algebra). The corresponding central idempotents $e_{1}, \dots, e_{t}$ give the block decomposition $B_{i} = e_{i} k G$ . Since distinct blocks are orthogonal ( $B_{i} B_{j} = 0$ for $i \neq = j$ ), the projective indecomposables belonging to different blocks are disjoint, and the Cartan matrix is block diagonal. $□$

Connections [Master]

Maschke's theorem 07.02.01. The failure of Maschke in characteristic $p$ dividing $∣ G ∣$ is the starting point for the cde-triangle. When Maschke holds, $R_{k} (G)$ is free on irreducibles and the Cartan matrix is the identity. The cde-triangle measures the deviation from this semisimple ideal, with the Cartan invariants quantifying the failure of complete reducibility.
Brauer induction theorem 07.01.11. Brauer induction builds $R (G)$ as the $Z$ -span of characters induced from elementary subgroups. Combined with the decomposition map, this gives a construction of $R_{k} (G)$ from local data at each prime dividing $∣ G ∣$ , connecting the global representation ring to the local modular theory.
Character orthogonality 07.01.04. The ordinary orthogonality relations extend to the modular setting: Brauer characters satisfy orthogonality on $p$ -regular elements, and the Cartan matrix $C = D^{T} D$ is the Gram matrix of this modular inner product. This is the foundation for the modular character theory developed in 07.02.04.
Non-abelian Fourier transform 07.01.09. The Fourier transform on $G$ decomposes functions into characters, organised by $R (G)$ . The cde-triangle extends this decomposition to the modular setting, where the Fourier analysis is governed by the Cartan matrix rather than the character table alone.

Historical & philosophical context [Master]

Alexander Grothendieck introduced the Grothendieck group in his 1957 paper Sur quelques points d'algebre homologique (Tohoku Math. J. 9) ^{[Grothendieck1957]}, constructing $K_{0}$ as the universal recipient of an Euler-characteristic map from an exact category. Richard Brauer had already developed the cde-triangle in his 1956 paper Zur Darstellungstheorie der Gruppen endlicher Ordnung (Math. Z. 63) ^[Brauer1956], establishing the relation between ordinary and modular characters via the decomposition and Cartan matrices without the K-theoretic language. The synthesis of Brauer's triangle with Grothendieck's framework crystallised in the 1960s through Curtis and Reiner's Representation Theory of Finite Groups and Associative Algebras ^{[CurtisReiner1962]}, which provided the definitive treatment.

The Adams operations on $R (G)$ , introduced by J. F. Adams in his 1962--66 sequence of papers on the $J$ -homomorphism (Topology 2--5), connect the representation ring to stable homotopy theory and appear in the Atiyah-Segal completion theorem. The deeper connection between the cde-triangle and algebraic K-theory --- via the Cartan map as the $K_{0}$ -level of a map of spectra --- is developed in Swan's 1970 monograph K-Theory of Finite Groups and Orders (Springer LNM 149) ^[Swan1970].

Bibliography [Master]

@article{Grothendieck1957,
  author = {Grothendieck, Alexander},
  title = {Sur quelques points d'algebre homologique},
  journal = {Tohoku Math. J.},
  volume = {9},
  year = {1957},
  pages = {119--221},
}

@article{Brauer1956,
  author = {Brauer, Richard},
  title = {Zur Darstellungstheorie der Gruppen endlicher Ordnung},
  journal = {Math. Z.},
  volume = {63},
  year = {1956},
  pages = {406--444},
}

@book{CurtisReiner1962,
  author = {Curtis, Charles W. and Reiner, Irving},
  title = {Representation Theory of Finite Groups and Associative Algebras},
  publisher = {Wiley},
  year = {1962},
}

@book{Serre1977,
  author = {Serre, Jean-Pierre},
  title = {Linear Representations of Finite Groups},
  publisher = {Springer},
  year = {1977},
}

@book{LangAlgebra,
  author = {Lang, Serge},
  title = {Algebra},
  publisher = {Springer},
  year = {2002},
  edition = {3rd},
}

@book{Swan1970,
  author = {Swan, Richard G.},
  title = {K-Theory of Finite Groups and Orders},
  publisher = {Springer},
  year = {1970},
  series = {Lecture Notes in Mathematics},
  volume = {149},
}

Prerequisites

07.01.09
07.01.10
07.01.11
07.01.12

Tier anchors

beginner: Serre Part III informal; the accounting ledger for representations
intermediate: Serre Part III Sections 12-15; Curtis-Reiner Vol. II Section 16C
master: Grothendieck 1957 Tohoku Math. J. 9; Brauer 1956 Math. Z. 63; Serre Linear Representations Part III; Curtis-Reiner Methods of Representation Theory Vol. I

References

TODO_REF
Grothendieck — Sur quelques points d'algebre homologique · Tohoku Math. J. 9 (1957), Section 2
TODO_REF
Brauer — Zur Darstellungstheorie der Gruppen endlicher Ordnung · Math. Z. 63 (1956)
TODO_REF
Serre — Linear Representations of Finite Groups · Part III, Sections 12-15
TODO_REF
Curtis-Reiner — Methods of Representation Theory Vol. I · Section 16C, decomposition and Cartan matrices
TODO_REF
Lang — Algebra · Ch. XVIII, representation rings
TODO_REF
Swan — K-Theory of Finite Groups and Orders · Springer LNM 149 (1970)

Reviewer

TBD

Estimated time

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