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Modular representation theory

Modular representation theory is a branch of representation theory that examines linear representations of finite groups over fields of positive characteristic p, especially when p divides the order of the group, leading to non-semisimple structures unlike the completely reducible representations in characteristic zero.^[1]^[2] In this setting, a representation is a homomorphism from the group to the general linear group over a vector space, but Maschke's theorem fails to guarantee complete reducibility, resulting in indecomposable modules that cannot be expressed as direct sums of irreducibles.^[1] Key concepts include irreducible modules, which admit no nontrivial submodules, and projective modules, which play a central role in resolving the complexities arising from the characteristic dividing the group order.^[2] Tools such as Brauer characters, defined on p-regular elements of the group, extend ordinary character theory to identify composition factors and decomposition numbers, while block theory decomposes the group algebra into indecomposable components linked by central characters.^[1]^[2] The Jacobson radical of the group algebra further aids in analyzing these structures by intersecting maximal ideals.^[1] Historically, the foundations were laid by Maschke's work in 1899 on semisimple representations, but modular theory advanced significantly through Richard Brauer's contributions starting in 1935, including character-theoretic methods that influenced the classification of finite simple groups.^[1]^[2]^[3] Later developments by J.A. Green introduced module-theoretic approaches, emphasizing rings and algebras, with applications extending to symmetric groups, Lie-type groups, and connections to quantum groups and diagrammatic algebras like the Temperley-Lieb algebra.^[2] These ideas underpin broader areas in algebra, including the study of Cartan matrices and Grothendieck groups for tracking module compositions.^[2]

Historical Development

Origins and Early Work

Modular representation theory emerged in the late 19th century as mathematicians sought to extend the theory of linear representations of finite groups from fields of characteristic zero, such as the complex numbers, to fields of positive characteristic. A foundational contribution came from Richard Dedekind in 1897, who examined the decomposition of the regular representation of the symmetric group S_3 over fields of characteristic 2 and 3 in his supplements to Dirichlet's Vorlesungen über Zahlentheorie. For characteristic 2, Dedekind computed the group determinant \Theta(S_3) and observed that it factors as (\Phi_1 \Phi_3)^2 \mod 2, where \Phi_3 is an irreducible quadratic factor appearing with multiplicity 2, exceeding its degree, indicating non-semisimplicity. Similarly, in characteristic 3, \Theta(S_3) \equiv (\Phi_1 \Phi_2)^3 \mod 3, with factors appearing to multiplicity 3. These explicit computations highlighted deviations from characteristic-zero behavior, laying groundwork for understanding modular decompositions.^[4] Issai Schur built upon these ideas in the early 1900s, extending representation theory to symmetric groups S_n and incorporating initial modular considerations. In his 1901 doctoral thesis and subsequent 1905 paper, Schur developed a comprehensive framework for the irreducible representations of S_n over the complex numbers, using Young tableaux to parametrize them, but he also explored integral forms and reductions modulo primes. These efforts revealed how ordinary representations of symmetric groups behave under modular reduction, particularly when the characteristic divides the group order, influencing later modular classifications. Schur's work connected group representations to symmetric polynomials and invariant theory, providing tools for analyzing modular cases through combinatorial methods.^[5] A key motivation for modular theory arose from the failure of Maschke's theorem in positive characteristic, first articulated by Heinrich Maschke in 1898 for characteristic zero, where group algebras are semisimple. Ferdinand Georg Frobenius, in his 1903 paper on linear substitutions and bilinear forms, proved that the group algebra \mathbb{F}[G] over a field \mathbb{F} of characteristic not dividing |G| is semisimple, but fails otherwise, as the averaging projector no longer works due to division by |G| becoming impossible. This semisimple structure underpinned ordinary representation theory but broke down modularly, prompting investigations into indecomposable representations and blocks. Early 20th-century developments, notably by Leonard Eugene Dickson in his 1907 address on modular theory of group characters, linked these issues to invariant theory and modular class functions. Dickson extended Frobenius's character orthogonality to prime characteristic, using class functions to study reductions, and connected modular representations to invariants of binary forms under modular transformations, bridging algebra and number theory.^[6]^[4]

Key Advances and Modern Contributions

Richard Brauer's foundational work in the 1930s and 1950s established the framework for modular character theory and block decomposition, enabling the study of representations over fields of characteristic dividing the group order. In particular, his 1941 paper introduced key relations between ordinary and modular characters, culminating in the theorem that the number of irreducible ordinary characters in a block equals the number of irreducible modular characters in that block.^[7] This result, often referred to as the Brauer-Cartan theorem in this context, bounds the number of simple modules per block and underpins subsequent block theory. Brauer characters, developed during this era as traces of modular representations on p-regular elements, serve as essential tools for lifting ordinary characters to modular settings.^[7] In the 1950s, James A. Green advanced the local structure of modules by introducing vertices and sources for indecomposable modules, providing a way to associate p-subgroups to module projectivity. Green's 1959 work defined the vertex of an indecomposable kG-module as a minimal p-subgroup Q such that the module is projective relative to N_G(Q), with sources capturing the local behavior over the normalizer. This framework, formalized in his Green correspondence, links indecomposable modules across subgroups and has become central to analyzing module lattices.^[8] During the 1950s and 1960s, contributions from Hisao Nagao and others refined defect groups and block invariants, shifting focus toward p-local properties. Nagao's 1962 theorem provided a module-theoretic analogue to Brauer's second main theorem, relating block idempotents to defect group actions.^[9] These developments solidified defect groups as conjugacy classes of p-subgroups determining block multiplicity and fusion, with applications to symmetric and alternating groups.^[10] Post-1980 extensions to finite groups of Lie type have emphasized block invariants and equivalences, notably through work by Michel Broué and Jon Alperin. Broué's 1980s conjectures on abelian defect groups for principal blocks of Lie type groups link modular representations to affine Weyl groups via derived equivalences.^[11] Alperin's fusion theorem (1986) and joint results with Broué classify block invariants like the number of simple modules via p-local data, facilitating computations for groups like GL_n(q).^[12] These invariants have proven crucial for verifying Brauer's k(B)-conjecture in Lie type settings.^[13] The Alperin-McKay conjecture, proposed in the 1980s, posits that for a prime p, the number of irreducible characters of degree not divisible by p equals that for p-subgroups, with block-wise versions refining fusion patterns; it remains open in general as of 2025, though a 2025 result completes its proof for the prime 2 in quasi-isolated blocks of exceptional groups of Lie type, alongside partial resolutions for maximal defect blocks.^[14] Recent progress includes inductive verifications for quasi-isolated blocks, reducing it to local conditions.^[15] Computational tools have transformed modular representation theory, addressing gaps in manual verification; MAGMA's implementation of the MeatAxe algorithm decomposes modules over finite fields to compute Brauer characters and decomposition matrices.^[16] The MeatAxe, integrated into GAP and standalone, has facilitated computations for sporadic groups and symmetric groups, enabling checks of block invariants.^[17]^[18]

Basic Concepts and Examples

Definition and Setup

Modular representation theory is the study of representations of finite groups over fields of positive characteristic. For a finite group G and a field k of characteristic p > 0 dividing the order |G|, a modular representation of G over k is a finite-dimensional kG-module, where kG denotes the group algebra of G over k.^[1] This framework contrasts sharply with ordinary representation theory over fields of characteristic zero, such as \mathbb{C}, where every representation is semisimple (completely reducible into a direct sum of irreducible representations) by Maschke's theorem, as the group order |G| is invertible in the field.^[1]^[19] In the modular setting, semisimplicity fails because p divides |G|, rendering the averaging operator over the group elements noninvertible in k.^[19] Consequently, [kG](/page/KG)-modules are generally indecomposable and exhibit more complex structure, with every finite-dimensional module possessing a composition series whose factors are simple modules.^[1] The algebra kG itself is Artinian (as a finite-dimensional algebra over a field) but not semisimple, leading to the study of its Jacobson radical and related invariants to understand module categories.^[1] The setup typically assumes k is algebraically closed of characteristic p, ensuring that every irreducible representation appears in a completely reducible module over an extension; more generally, k may be any splitting field for kG, meaning the algebra decomposes into a direct product of matrix algebras over division rings that split over k.^[19] The term "modular" specifically denotes representations in characteristic p dividing |G|, distinguishing it from the broader "characteristic p" context where p may not divide the group order, and reflects the origins in modular arithmetic modulo p.^[1] This terminology evolved in the early 20th century alongside the development of the theory, emphasizing the reduction modulo p from characteristic zero cases.^[20]

Illustrative Example

A concrete illustration of modular representation theory arises from the symmetric group S_3, which has order 6 and presentation \langle \sigma, \tau \mid \sigma^3 = \tau^2 = 1, \tau \sigma \tau = \sigma^{-1} \rangle where \sigma = (1\,2\,3) and \tau = (1\,2). Consider the group algebra kS_3 over the field k = \mathbb{F}_2 of characteristic 2. This algebra has dimension 6 with standard basis \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau\}.^[21] The augmentation map is the k-linear trace \varepsilon: kS_3 \to k defined by \varepsilon\left( \sum_{g \in S_3} a_g g \right) = \sum_{g \in S_3} a_g, which is a surjective algebra homomorphism. The augmentation ideal is I = \ker \varepsilon = \operatorname{span}_k \{ g + 1 \mid g \in S_3, g \neq 1 \} (noting that -1 = 1 in characteristic 2), which has dimension 5 and coincides with the Jacobson radical \operatorname{rad}(kS_3). The quotient kS_3 / I \cong k realizes the trivial representation as a simple module.^[22]^[21] In characteristic 2, S_3 has two irreducible representations up to isomorphism: the 1-dimensional trivial module D^{(3)} (where the superscript denotes the partition labeling the Specht module) and the 2-dimensional simple module D^{(2,1)}. The latter admits an explicit basis \{e_1, e_2\} where e_1 = \{1\,2\,3\} + \{3\,2\,1\} and e_2 = \{1\,3\,2\} + \{2\,3\,1\} in the permutation basis, with action \sigma \cdot e_1 = e_2 and \sigma \cdot e_2 = e_1 + e_2, while transpositions act by swapping or fixing accordingly. The regular module kS_3 decomposes into two blocks: the principal block (spanned by the idempotent e_1 = 1 + \sigma + \sigma^2) containing the trivial simple, and a unipotent block (spanned by e_2 = \sigma + \sigma^2) containing the 2-dimensional simple.^[23]^[21] Non-semisimplicity is evident in the permutation module M = k \oplus k \oplus k with basis \{e_1, e_2, e_3\} corresponding to the standard action of S_3 on three points. The subspace U = \langle e_1 + e_2 + e_3 \rangle is the 1-dimensional trivial socle of M, and the quotient M/U \cong D^{(2,1)} is the 2-dimensional simple head. Thus, M is an indecomposable module of Loewy length 2 with composition factors D^{(3)} (multiplicity 1) and D^{(2,1)} (multiplicity 1), realizing a non-split extension $0 \to D^{(3)} \to M \to D^{(2,1)} \to 0. The submodule lattice of M is a chain: \{0\} \subset U \subset M with successive quotients U/\{0\} \cong D^{(3)} and M/U \cong D^{(2,1)}.^[22]^[21] In characteristic 3 over k = \mathbb{F}_3, the irreducibles are the 1-dimensional trivial D^{(3)} and sign D^{(1,1,1)} modules (the latter nontrivial since the sign is faithful in odd characteristic not dividing 3). The group algebra kS_3 remains non-semisimple, with both indecomposable projectives of dimension 3 having Loewy length 3 and composition factors mixing the trivial and sign modules (Cartan matrix \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}). For instance, the projective cover of the trivial has series with factors trivial (socle), sign, trivial (head). The 2-dimensional representation from characteristic 0 reduces modulo 3 to the semisimple module D^{(3)} \oplus D^{(1,1,1)}.^[22]^[21]

Algebraic Foundations

Group Algebra over Modular Rings

The group algebra kG of a finite group G over a field k of characteristic p is the associative k-algebra consisting of all formal k-linear combinations \sum_{g \in G} c_g g with c_g \in k, equipped with multiplication extended linearly from the group operation via g \cdot h = gh for g, h \in G.^[24] This endows kG with a basis \{g \mid g \in G\} of cardinality |G|, making it a finite-dimensional algebra of dimension |G| over k.^[24] The unit element is the identity $1_G of G, and the algebra is unital.^[24] As a finite-dimensional algebra over a field, kG is Artinian and possesses rich ring-theoretic structure, notably as a symmetric Frobenius algebra regardless of whether p divides |G|.^[24] The Frobenius form is the nondegenerate bilinear pairing \beta: kG \times kG \to k defined by \beta(a, b) as the coefficient of the identity element in the product a b, which satisfies \beta(a b, c) = \beta(a, b c) for all a, b, c \in kG.^[25] This symmetry follows from \beta(a, b) = \beta(b, a) since the coefficient extraction is invariant under reversal via inverses in G.^[24] Consequently, kG is quasi-Frobenius, meaning it is injective as a module over itself on both sides, with every projective module being injective and the socle and top composition factors isomorphic.^[26] In the broader context of ring theory, kG serves as a prototypical example of a finite-dimensional Hopf algebra, with coproduct \Delta(g) = g \otimes g, counit \epsilon(g) = 1, and antipode S(g) = g^{-1}, though its finite-dimensionality underscores its role in modular representation theory.^[24] The center Z(kG) of kG is the subalgebra of elements that commute with every basis element, and it has k-dimension equal to the number of conjugacy classes of G; a basis for Z(kG) is given by the class sums c_C = \sum_{g \in C} g over each conjugacy class C of G.^[21] This spanning property holds independently of the characteristic p, as conjugation preserves the linear independence of these sums.^[21] When p does not divide |G|, kG is semisimple by Maschke's theorem, and the Artin-Wedderburn theorem decomposes it as kG \cong \prod_{i=1}^l M_{n_i}(D_i), where each D_i is a finite-dimensional division algebra over k and the n_i are the dimensions of the irreducible representations.^[24] In the modular case where p divides |G|, kG is indecomposable as an algebra but possesses a Jacobson radical J(kG), and the quotient kG / J(kG) is semisimple with an analogous Artin-Wedderburn decomposition into matrix algebras over division rings, where the division rings may be non-commutative extensions adapted to the characteristic p unless k is a splitting field.^[24] Simple kG-modules in the semisimple case correspond to primitive central idempotents in Z(kG).^[21]

Reduction Modulo p

In modular representation theory, the process of reducing ordinary representations over the complex numbers \mathbb{C} to modular representations over a field k of characteristic p begins with an integral model. Specifically, consider a \mathbb{C}G-representation V realized via a \mathbb{Z}G-lattice L, which is a free \mathbb{Z}-module of finite rank equipped with a G-action compatible with the group ring \mathbb{Z}G. The reduction modulo p yields the kG-module \bar{L} = L/pL \otimes_{\mathbb{F}_p} k, where the tensor product ensures the structure over the splitting field k. This construction bridges characteristic zero and positive characteristic, allowing the study of modular structure through ordinary data.^[27] The isomorphism class of \bar{L} depends on the choice of lattice L, as different \mathbb{Z}-forms of the same V can produce non-isomorphic modular modules. However, all such reductions share the same composition factors, meaning they have identical Jordan-Hölder multiplicities for the simple kG-modules. This property follows from the consistency of Brauer characters across equivalent integral forms, ensuring that the modular content is invariant under lattice selection.^[27] For a \mathbb{C}G-representation \rho with character \chi, the modular reduction is captured by specializing the character values modulo p, but this requires embedding the cyclotomic field containing \chi(g) into a p-adic completion and reducing via the maximal ideal. Formally, if \chi(g) lies in the ring of algebraic integers \overline{\mathbb{Z}}, the Brauer character \phi of \bar{\rho} on p-regular elements is given by \phi(g) = \sum \overline{\theta_i(g)}, where \theta_i are lifts of eigenvalues to characteristic zero, but the result is non-unique due to embedding choices and lattice variations. This non-canonical nature underscores the role of the decomposition matrix in relating ordinary and modular characters precisely.^[27] Brauer's lifting theorem guarantees that every simple kG-module S appears as a composition factor in \bar{L} for some irreducible \mathbb{Z}G-lattice L associated to an irreducible \mathbb{C}G-module. The theorem establishes the surjectivity of the reduction map on the level of Grothendieck groups, with decomposition numbers d_{\chi,S} \geq 0 integers recording multiplicities, and ensures no modular simple is "missed" in the ordinary-to-modular transition.^[27] Illustrative examples of this reduction process reveal indecomposable structures akin to Jordan blocks. For the symmetric group S_3 with p=3, the 2-dimensional irreducible ordinary representation reduces to a uniserial kG-module of length 2, with simple head (the sign module) and socle (the trivial module), demonstrating how non-semisimple extensions emerge modulo p. Similar reductions in dihedral groups or p-groups often yield chains of simple factors, highlighting the breakdown of complete reducibility in characteristic p.^[27] Post-2000 developments have refined this framework through p-adic lifts, allowing modular representations to be elevated to modules over p-adic rings like \mathbb{Z}_p or \mathbb{Z}/p^2\mathbb{Z} for greater precision in deformations. For instance, in the representation theory of \mathrm{SL}_2(p^r), basic homogeneous representations V_i(p^r) (for $1 \leq i \leq p) lift to \mathbb{Z}/p^2\mathbb{Z} if and only if r=1 and specific conditions on p and i hold, such as i = p-2 or p-1 for odd p, with further lifts to \mathbb{Q}_p possible; these results rely on computing Ext-groups to resolve obstructions.^[28]^[29] These reductions connect to stable isomorphism classes, where two kG-modules M and N (arising from different lattices) are stably isomorphic if M \oplus P \cong N \oplus Q for some projective modules P, Q. Since projectives are trivial in the stable category, this equivalence preserves essential modular invariants like composition factors and endomorphism rings up to stable structure, facilitating comparisons across lattice choices.^[27]

Character Theory

Brauer Characters

In modular representation theory, the Brauer character of a kG-module M, where k is a field of characteristic p and G is a finite group, is defined as a class function \phi_M: G_{p' } \to \mathbb{C} on the p-regular elements G_{p'}, taking values in a cyclotomic field. For a p-regular g \in G, \phi_M(g) is the sum \sum_i \theta(\lambda_i), where \lambda_1, \dots, \lambda_{\dim_k M} are the eigenvalues of the matrix representing the action of g on M (over an algebraic closure of k), and \theta: k^\times \to \mathbb{C}^\times is a fixed embedding sending nonzero elements of k to roots of unity of order prime to p.^[30]^[31]^[32] Brauer characters are additive: for modules M and N, \phi_{M \oplus N} = \phi_M + \phi_N, and more generally, they respect short exact sequences. The irreducible Brauer characters, corresponding to the simple kG-modules, form a basis for the space of class functions on G_{p'}.^[31]^[32]^[1] To compute a Brauer character, one lifts the modular representation to characteristic zero via a modular system and restricts to p-regular elements, or directly finds the eigenvalues modulo p and applies the embedding \theta to obtain the trace as \phi_M(g) = \sum_i \theta(\lambda_i).^[32]^[31]^[1] For cyclic groups, Brauer characters simplify due to the diagonalizability of representations. Consider G = C_3 = \langle x \mid x^3 = 1 \rangle and p=3; in characteristic zero, there are one-dimensional trivial and sign representations, but since p divides |G|, there is a unique irreducible of dimension 1 (the trivial). Here, G_{p'} = \{1\}, so Brauer characters are determined by their value at the identity, which equals the module dimension; for the regular module, \phi(1) = 3. (In general, the regular module has Brauer character |G| at 1 and 0 at other p-regular elements.) Recent work provides bounds on Brauer character degrees; for instance, if a prime q (odd, with (p,q) \ne (2,3)) divides the degree of every nonlinear irreducible p-Brauer character, then G has a normal q-complement.^[31]^[33]

Orthogonality Relations

The orthogonality relations for Brauer characters provide fundamental tools for decomposing modular representations, mirroring the role of Frobenius-Schur orthogonality in characteristic zero but restricted to p-regular elements of the finite group G. Let φ and ψ denote Brauer characters of FG-modules, where F is a splitting field of characteristic p. The inner product is defined as

\langle \phi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G_{p'}} \phi(g) \overline{\psi(g^{-1})},

where G_{p'} is the set of p-regular elements (those of order coprime to p), and the sum is over conjugacy classes weighted appropriately by centralizer sizes in the unnormalized form.^[21] This Hermitian form equips the space of F-class functions on G_{p'} with an orthonormal structure. For distinct irreducible Brauer characters φ_i and φ_j belonging to Irr_F(G), the basic row orthogonality relation states that ⟨φ_i, φ_j⟩ = δ_{ij}, ensuring linear independence and establishing that the set {φ_i} forms an orthonormal basis for the space of generalized Brauer characters.^[21] The unnormalized version follows as

\sum_{K \in Cl_{p'}(G)} \frac{\phi_i(g_K) \overline{\phi_j(g_K)}}{|C_G(g_K)|} = \delta_{ij},

where Cl_{p'}(G) denotes the p-regular conjugacy classes and g_K is a representative of class K; here, l = |Cl_{p'}(G)| is the number of such classes, and the relation highlights the p-part |G|_p in the normalization when projecting to block structures.^[31] Column orthogonality in the modular setting extends the classical induction formulas and involves projective indecomposable characters Φ_ψ associated to irreducible Brauer characters ψ. Specifically, for p-regular elements g and h,

\sum_{\psi \in \mathrm{Irr}_F(G)} \psi(g^{-1}) \Phi_\psi(h) = |C_G(g)| \delta_{g \sim h},

where the sum is over irreducibles and δ_{g \sim h} is 1 if g and h are conjugate, else 0; this holds for induction from subgroups via Brauer's induction theorem, allowing decomposition of induced Brauer characters from p-regular classes in subgroups.^[21] These relations imply symmetry properties, such as the unitarity of the Brauer character table when viewed as a matrix over p-regular classes, and facilitate computations of dimensions and multiplicities in the Grothendieck group of FG-modules.^[31] A key application arises in decomposing permutation characters modulo p. The Brauer character of a permutation FG-module, obtained by reducing the ordinary permutation character to characteristic p and restricting to p-regular elements, decomposes as ∑_i m_i φ_i, where the multiplicity m_i = ⟨φ_i, \mathrm{perm}^B⟩ equals the number of fixed points of the permutation on p-regular elements, averaged appropriately; this inner product yields explicit formulas for the modular constituents of transitive permutation representations.^[21] Generalized orthogonality relations refine these for characters within p-blocks of the group algebra, incorporating height-zero characters and defect groups. Broué established such relations, showing that characters of height zero in a block satisfy enhanced orthogonality with respect to block idempotents, bounding the number of irreducibles and linking to local structure.

Module Structure

Simple Modules and Their Number

In modular representation theory, the simple kG-modules, where G is a finite group and k is a field of characteristic p > 0, are the irreducible modules up to isomorphism, denoted D_1, \dots, D_l. The number l of these distinct simple modules equals the number of p-regular conjugacy classes in G, as established by Brauer's theorem.^[27] A conjugacy class is p-regular if its elements have order coprime to p. These simple modules are labeled by the irreducible Brauer characters, which are the characters afforded by the simple kG-modules evaluated on p-regular elements.^[32] Assumptions often include k being a splitting field, where the simple modules are absolutely irreducible, \operatorname{End}_{kG}(D_i) = k, and Brauer characters fully capture their traces on p-regular elements. In the structure of general finite kG-modules, the simple modules appear as composition factors; specifically, the head of a module is its maximal semisimple quotient, which is a direct sum of simple modules, and the socle is its maximal semisimple submodule, likewise a direct sum of simples. Over a splitting field k for kG, each simple module D_i satisfies \dim_k \operatorname{End}_{kG}(D_i) = 1, by the modular analogue of Schur's lemma, implying that the endomorphism ring is exactly k.^[32] In non-splitting fields, the endomorphism ring \operatorname{End}_{kG}(D_i) is a finite-dimensional division algebra over k, with dimension greater than 1, leading to more complex realization of the simples as representations.^[27] The composition multiplicities d_{ij} quantify how ordinary irreducible characters \chi_j decompose into modular simples upon reduction modulo p, defined as the multiplicity [S_j : D_i], where S_j is the simple \mathbb{C}G-module affording \chi_j. These multiplicities form the decomposition matrix, central to linking ordinary and modular theory.

Projective Modules

In modular representation theory, a kG-module P, where k is a field of characteristic p and G is a finite group with p dividing |G|, is projective if it is a direct summand of a free kG-module, equivalently if the functor Hom_kG(P, −) is exact. The group algebra kG itself is projective as the free kG-module of rank 1, and it decomposes as a direct sum of indecomposable projective modules P_i (i = 1, \dots, l(G)), where l(G) is the number of simple kG-modules; these P_i are unique up to isomorphism and form a complete set of representatives for the indecomposables. Each P_i has a simple head D_i = P_i / \mathrm{rad}_{kG}(P_i), establishing a bijection between the isomorphism classes of indecomposable projectives and simple modules. Every finite-length kG-module M admits a projective cover, a surjective kG-homomorphism π: Q → M from an indecomposable projective Q = P(D) with kernel \mathrm{rad}_{kG}(Q), unique up to isomorphism, such that any other surjection from a projective to M factors through π. This cover allows the construction of minimal projective resolutions, sequences \cdots \to P_1 \to P_0 \to M \to 0 where the P_j are indecomposables and the images of the maps generate the radicals, providing tools for computing \operatorname{Ext} groups and cohomological dimensions in the category of kG-modules. The dimension of the indecomposable projective P_i is given by \dim_k P_i = |G|_p \dim D_i, where |G|_p denotes the p-part of |G| (the highest power of p dividing |G|). The Green correspondence provides a bijection between the indecomposable kG-modules with vertex J and the indecomposable kN_G(J)-modules with vertex J, where J is a p-subgroup of G, such that for corresponding modules M and N, M is isomorphic to \operatorname{Ind}_{N_G(J)}^G N \oplus (projective kG-module), and conversely N is a direct summand of \operatorname{Res}_G^{N_G(J)} M up to projectives. This correspondence preserves the lattice of submodules and is essential for reducing the study of global module structure to local data near p-subgroups.^[34] For an indecomposable kG-module M, a vertex is a minimal p-subgroup Q ≤ G such that M is relatively Q-projective, meaning M is a direct summand of \mathrm{Ind}_Q^G N for some kQ-module N; all vertices are conjugate, and Q is essential in the sense that no proper subgroup of Q has this property. The corresponding source module N is indecomposable over kQ with M a direct summand of \mathrm{Ind}_Q^G N, and for projective M, the vertex is the trivial subgroup with source the trivial module. The vertices classify the "p-local" behavior of modules, linking global projectives to local sources over p-subgroups. Source modules play a key role in describing projectives via induction from p-subgroups, and the Endo-Levi theorem characterizes the endomorphism ring \operatorname{End}_{kG}(P) of an indecomposable projective P as a local ring with radical structure determined by the source, providing a decomposition into matrix rings over division rings with p-group action; originally proved in the 1950s using classical methods, modern proofs employ stable homotopy theory of the module category.

Advanced Block Theory

Blocks of the Group Algebra

In modular representation theory, the group algebra kG over an algebraically closed field k of characteristic p > 0 decomposes as a direct sum kG = \bigoplus_b b kG, where the sum runs over the primitive central idempotents b in the center Z(kG). Each such b determines a block B = b kG, which is a two-sided ideal of kG and serves as the identity element for modules in that block. The simple kG-modules are partitioned into these blocks, with a module M belonging to the block b if bM = M. This decomposition arises from the semisimple structure of the commutative ring Z(kG), whose dimension equals the number of blocks.^[32]^[35] Brauer's block theory establishes a correspondence between blocks of kG and certain subsets of ordinary irreducible characters and modular irreducible (Brauer) characters, linked through the decomposition matrix D. Specifically, the matrix D, whose entries are the multiplicities of modular simples in the reductions modulo p of ordinary characters, takes a block-diagonal form with respect to this partition: D = \operatorname{diag}(D_{B_1}, \dots, D_{B_t}), where each D_B describes the linkages within block B. This framework reveals how blocks encode the interaction between characteristic-zero and modular representations, with ordinary characters in a block B decomposing into modular characters also in B.^[35]^[36] The number of blocks is at most the number of p-regular conjugacy classes in G, as the latter equals the number of irreducible Brauer characters (by the Brauer-Nesbitt theorem), and each block contains at least one such character. Each block b is associated with a central character \omega_b, a linear functional on the space of class functions on p-regular elements, defined by \omega_b(\sum g \in Cl_G(x)) = \operatorname{trace}(b \sum g) for p-regular x \in G, up to scalar multiple. These central characters distinguish the blocks and extend the trace form restricted to p-regular elements.^[32]^[35] Locally, each block b kG is indecomposable as a kG-bimodule, meaning it cannot be expressed as a nontrivial direct sum of bimodules. This indecomposability reflects the block's role in localizing the module category and underpins further structures like fusion systems, which model p-subgroup interactions within the block (as developed in works post-2000, e.g., Ragnarsson's contributions on block fusion systems).^[36]^[35]

Decomposition and Cartan Matrices

In modular representation theory, the decomposition matrix D relates the irreducible ordinary characters of a finite group G to its irreducible Brauer characters in characteristic p. The rows of D are indexed by the ordinary irreducible characters \chi_i \in \operatorname{Irr}(G), while the columns are indexed by the irreducible Brauer characters \phi_j \in \operatorname{IBr}_p(G). The entry d_{ij} is the multiplicity with which the simple kG-module affording \phi_j appears as a composition factor in the reduction modulo p of the KG-module affording \chi_i, where K is a field of characteristic zero and k is its residue field of characteristic p.^[27]^[32] The matrix D has non-negative integer entries and is independent of the choice of modular system, provided it is a splitting field. On p-regular elements of G, the ordinary character satisfies \chi_i = \sum_j d_{ij} \phi_j. The 0-1 conjecture posits that all entries of D are 0 or 1, but this remains unproven in general; computational verifications confirm it holds for many small groups and certain classes, such as symmetric groups up to degree 17 in characteristic 2, though larger cases suggest potential complexity without known counterexamples as of 2025.^[27]^[35] The matrix D decomposes into block-diagonal form corresponding to the p-blocks of G, with each block submatrix having full column rank equal to the number of Brauer characters in that block.^[32] The Cartan matrix C encodes the composition structure of the projective indecomposable kG-modules. Its entries c_{ij} are defined as the dimension of \operatorname{Hom}_{kG}(P_j, P_i), where P_j is the projective cover of the simple module with Brauer character \phi_j, or equivalently, the multiplicity of the simple head of P_i (isomorphic to the socle of P_j) in the composition series of P_j / \operatorname{rad}(P_j). When k is a splitting field for G, C is symmetric and positive definite, with determinant a power of p, and satisfies the key relation C = D^T D. This implies c_{ij} = \sum_l d_{li} d_{lj}, linking the multiplicities in projective modules to those in ordinary reductions.^[27]^[32] For the symmetric group S_3 in characteristic p=3, the decomposition matrix is

D = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{pmatrix},

with ordinary characters \chi_1 (trivial), \chi_2 (sign), and \chi_3 (standard), and Brauer characters \phi_1, \phi_2. The corresponding Cartan matrix is

C = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix},

verifying C = D^T D and showing each projective indecomposable has two simple composition factors. In characteristic p=2 for S_3, D = \begin{pmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{pmatrix}, yielding C = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}. These examples illustrate how D and C capture the non-semisimple structure when p divides |G|.^[32]^[27]

Defect Groups

In modular representation theory, a defect group of a p-block b of the group algebra kG, where G is a finite group and k is an algebraically closed field of characteristic p, is a maximal p-subgroup D of G such that b lies in the image of the induction map from kD to kG, denoted kD^G. This definition, introduced by Richard Brauer in 1959, ensures that defect groups are unique up to conjugation in G. By the Brauer correspondence theorem, the block b of G corresponds to a unique block b' of the normalizer N = N_G(D) with the same defect group D, and the number of simple modules satisfies l(b) = l(b'). In cases where D is normal in N and b' is the principal block of N, l(b) equals the number of irreducible k-representations of N/D. The size of the defect group encodes essential information about the block's structure, with larger defect groups corresponding to blocks of greater complexity. The defect number of the block b, denoted \operatorname{def}(b), is defined as the p-adic valuation v_p(|G : D|), measuring how close |D| is to the full p-part of |G|. For the principal block, which contains the trivial representation, the defect group is a Sylow p-subgroup of G, so \operatorname{def}(b) = v_p(|G|). Blocks sharing the same defect group (up to conjugacy) are Brauer equivalent, meaning there is a bijection between their ordinary characters and Brauer characters that preserves decomposition numbers. Moreover, the defect group governs the fusion of p-elements within the block, linking to the control of fusion systems via the normalizer N_G(D). Blocks with defect group of order p (defect number 1) are nilpotent, characterized by having exactly one simple module and indecomposable projectives that are induced from projective modules over the normalizer of the defect group. The restriction of any projective indecomposable module in b to D is a multiple of the regular kD-module. Further applications involve Alperin's fusion theorem, which asserts that every fusion of p-subgroups occurring in the block is realized within N_G(D), providing a local control mechanism for the block's structure. Ongoing research includes weight conjectures, such as Alperin's weight conjecture, positing that the number of simple modules in b equals the number of weights—pairs (Q, \pi) where Q is a p-subgroup of D and \pi is a p-defect zero character of N_G(Q)/Q—and this remains unresolved for arbitrary finite groups as of 2025.

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