BASIC MODULAR REPRESENTATION THEORY GYUJIN OH Contents 1. Algebra 1 1.1. Related randos 2 2. cde triangle 2 3. Brauer characters 4 4. Green correspondence 5 5. Block and defect 7 6. Example: SL2(Fp) 9 6.1. Brauer tree 10 7. Ext 10 8. Auslander-Reiten theory 11 I scratched the surface of Schneider’s Modular representation theory of nite groups. 1. Algebra Here our base ring is unital and our modules are left modules. ∙ Any artinian and noetherian module has Jordan-Holder composition series. ∙ A semisimple module is a direct sum of simple modules. Every module has a unique maximal semisimple submodule which we call as socle. It is just the sum of all simple submodules. ∙ Radical is intersection of maximal submodules. So that the radical of base ring is the usual Jacobson radical. Semisimple = artinian + zero radical. ∙ If M is artinian and f ∈ EndR(M) is injective, then f is bijective (Fitting’s lemma). Same for noetherian/surjective. This uses increasing/decreasing chain of im(f n) and ker(f n). In particular this shows that, for any indecomposable finite length R, EndR(M) is local. ∙ Given any idempotent e ∈ R, eRe ⊂ R is a subring with unit element e. Decomposing e = e1 + ⋯ + er as sum of pairwise orthogonal idempotents is the same as decomposing Re = Re1 ⊕ ⋯ ⊕ Rer into sum of nonzero left ideals. Thus Re is indecomposable i e is primitive. ∙ If R is left noetherian, R has only nitely many central idempotents, primitive central idempotents are pairwise orthogonal, and they sum up to 1. Given a central primitive idempotent e, M is said to belong to e-block if eM = M. Then not only eM = M, but ex = x for all x ∈ M. 1 ∙ For a 2-sided ideal I ⊂ R such that either every element in I is (topologically) nilpotent. Then, any idempotent of R/I lifts to an idempotent of R. This operation can be done preserving primitivity and orthogonality. ∙ A denition of projective module is that P such that HomR(P, −) is an exact functor. A module hom f ∶ M → N is essential if it’s surjective but f (L) ≠ N for any proper L ⊂ M. A projective cover of M is an essential R-module hom f ∶ P → M where P is projective. There is at most one projective cover up to isomorphism. ∙ Conversely if R is complete and R/ Jac(R) is left artinian, then any fg module has projective cover, which mod Jac(R) is an iso. ∙ The Grothendieck group R(A) of category of nite length A-modules is the same as the free ℤ-module generated over simple A-modules (denoted as A). The Grothendieck group K0(A) of category of nitely generated projective A-modules is generated over ℤ by nitely generated indecomposable projective A-modules (denoted as A), if A is left noetherian. ∙ In particular, if A is semisimple, then any A-module is projective. ∙ If A is left artinian, then K0(A) is freely generated by A. Thus we can consider the com- posed homomorphism cA ∶ ℤ[A] → ℤ[A] called the Cartan homomorphism. ∙ If A is complete and A = A/ Jac(A) is left artinian, then Grothendieck groups for A and A are the same. So, A → A/ Jac(A) is a projective cover, and A = P1 ⊕ ⋯ ⊕ Pr , a decomposi- tion into a sum of nitely generated indecomposable A-modules, and every such module occurs as direct summand. ∙ So if A is left artinian, we have A = {P1, ⋯ ,Pt }, and thus A = {M1, ⋯ ,Mt } where Mi = Pi/ Jac(A)Pi. 1.1. Related randos. ∙ A Frobenius algebra is a nite-dimensional algebra over a eld with a nondegenerate symmetric bilinear form such that (xy, z) = (x, yz). In particular k[G] is a Frobenius algebra via (x, y) ↦ xy(1) (the coecient of 1 of xy). This I think is also referred as symmetric Frobenius algebra at some places. ∙ A module is projective and indecomposable i P/ rad(P) = soc(P). ∙ Given a ring hom f ∶ A → B, a B-module P is relatively projective wrt f if, for any commutative diagram P 0 ~ M / N where , are B-module hom’s while 0 is an A-module hom, there is a B-module hom ∶ P → M replacing 0 still making the diagram commute. ∙ For any A-module L0, B ⊗A L0 is relatively projective. More generally P is relatively pro- jective i it is a direct summand of B ⊗A L0 for some A-module L0. ∙ There is criterion for left R[G]-module to be relatively R[H]-projective (for H ≤ G, R any commutative ring). 2. cde triangle We do modular representation theory. Let R be a (0, p)-ring for k, where k is an algebraically closed eld of characteristic p > 0, R is a complete local commutative domain such that mR is 2 principal, residue eld is k and K = Frac(R) is of characteristic zero. Then there is the ramication e index e ≥ 1 such that Rp = mR. The trinity of the cde triangle is ∙ RK (G) ∶= R(K[G]) = K0(K[G]), ∙ Rk(G) ∶= R(k[G]), ∙ K0(k[G]). Here G is a nite group. ∙ The c. This is the Cartan homomorphism cG ∶ K0(k[G]) → Rk(G). This makes sense because k[G] is a nite-dimensional k-vector space so of course this is left artinian. ∙ The d. This map dG ∶ RK (G) → Rk(G) is something like “mod p reduction.” Of course as usual this is not a priori well-dened. Namely, one chooses a G-invariant lattice inside a G-representation over K (which is possible b/c of averaging) and then reduce mod p (or R), but the resulting representation might not be isomorphic. But considering Rk(G) is like we’re in a “semisimplied world” and in particular the resulting module in Rk(G) is always the same. This is almost the same as Brauer-Nesbitt theorem. This particular statement can be proved by passing to a common renement (after mutiplying by suciently high power of R). – Perhaps a more classical statement of Brauer-Nesbitt is that a semisimple represen- tation of G over k is completely determined by characteristic polynomials of g ∈ G’s. ¤ ¤ ¡ ∙ The e. By the Jacobson radical theory we did before, K0(R[G]) ≅ ℤ[R[G]], R[G] = k[G] ¡ and K0(k[G]) ≅ ℤ[k[G]]. Thus we can dene eG ∶ K0(k[G]) → RK (G) by ∼ K0(k[G]) ←ãããããããã K0(R[G]) → K0(K[G]). Thus the triangle is of form dG RK (G) / Rk(G) e 9 eG cG K0(k[G]) Quite obviously the triangle commutes. ∙ If p does not divide |G|, then cde are all isomorphisms. ∙ If G is a p-group, then Jac(k[G]) = Ik[G], the augmentation ideal. In particular, k[G] is a local ring, and the trivial module is the only simple k[G]-module (“p-group lemma” says that there is always a xed vector). Thus Rk(G) and K0(k[G]) are both ℤ, and the diagram becomes [V ]↦dimK V RK (G) / ℤ b ? 1↦K[G] ⋅|G| ℤ I am not doing Cliord theory. ∙ Induction gives a well-dened map G indH ∶ RF (H) → RF (G), [W ] ↦ [F[G] ⊗F[H] W ], 3 for any eld F and H ≤ G. This is not a homomorphism but rather the image is an ideal in RF (G). ∙ Induction commutes with d. ∙ Brauer induction theorem says that G Rk(G) = ∑ indH (Rk(H)), H∈He where He is the set of elementary subgroups of G. From this one sees that d is surjective. m m ∙ c is injective and has nite cokernel. Furthermore p Rk(G) ⊂ im(c), where p ‖|G|. Proof uses Brauer induction to reduce to case of elementary subgroups. ∙ e is injective and its image is a direct summand of RK (G). This uses two bilinear forms ([V ], [W ]) ↦ dimK HomK[G](V,W ) RK (G) × RK (G) ããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããã→ ℤ, ([P], [V ]) ↦ dimk Homk[G](P,V ) K0(k[G]) × Rk(G) ããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããããã→ ℤ. The second bilinear form is perfect, and the rst one is also perfect if K is a splitting eld for G. ∙ Furthermore this duality shows that, over suciently large eld, so that K0(k[G]) and Rk(G) are free over ℤ, the matrices representing d and e are transposes to each other. Therefore, the Cartan matrix corresponding to the natural bases is symmetric! 3. Brauer characters ∙ Usual charcter theory gives a homomorphism Tr ∶ Rk(G) → Cl(G, k), where Cl(G, k) means the space of class functions. By Artin-Wedderburn, Tr ∶ k ⊗ℤ Rk(G) → Cl(G, k) is injective. This is why in characteristic p the usual character theory is less useful; you lose information by k⊗ℤ. We use V as the character of V . ∙ Somehow quite surprisingly there is the notion of p-regular(∼semisimple)/p-unipotent elements (e.g. there is Jordan decomposition), and the general slogan is that mod p repre- sentation theory is encoded not in all conjugacy classes but rather in p-regular conjugacy classes. ∙ To be more precise, a p-regular conjugacy class is a conj. class whose elements have order prime to p. An element is p-unipotent if it is of order a power of p. Then, there is Jordan decomposition, namely for any g ∈ G, there are unique gr , gu ∈ G such that gr is p- regular, gu is p-unipotent, g = gr gu = gugr . The construction is quite simple, namely you just take appropriate powers of g.