D ade's conjectures include proving the variance of the alternating sum of a certain G-stable function over the p-group complex of a finite group G. We also show how to use the topological properties of p-group complexes to reduce the alternating sums. Among other things, we also get a formula that expresses the McKay numbers of finite unitary groups in terms of partitions of integers.
More explicitly, it involves the proof of the annihilation of the alternating sum of a certain G-stable function over the complex p-group G. In Chapter 3, we give a reformulation of the conjecture for finite groups of Lie type.
Preliminary Lemmas
The Conjecture
Notation and Preliminary Lemmas
If N \ ^ N 2 and V = N 2/N i, we denote by Irr°(G, V ) the set of irreducible characters of G whose restriction to N 2 (and thus to V) is trivial, and denote by Irr1 (G, V ) the set of irreducible characters of G whose restriction to N \ is trivial but whose restriction to N 2 is non-trivial. In the above notation, we replace Irr with k to denote the size of the corresponding set of characters. Similarly, if q is a power of p, the g-height of ip is the exponent of q in the prime factorization of cp( 1).
By definition, Irr(G, X ) consists of the set of irreducible characters of G that span a part of X . 1) There is a 1-1 correspondence between IrrfP , r) and Irr(G, r) given by the inductive characters . Recall that kd(Gx, X i, pi) counts the number of irreducible characters of Gx that span any character in X i as well as pi whose height is d.
Partition of Integers
We define a function a), where / / i s partition and a is an element of If [x is as written above, then.
Chapter 3 Refinement and General Discussion
- A Refinement for The Finite Groups of Lie Type
- Strategy
- The General Linear and Unitary Groups
In part (1) of Lemma 3.1.3, the p-blocks of G are of positive defect axis of full defect and parameterized by the central characters of G. We consider the g-height of characters of G rather than a p-height to simplify notation. Cancellation occurs when we count characters in Irr1 (Pj-, W, p) for those P j that have a common internal modulus W appeared as a factor in its lower central row, so we only need to count the number of irreducible characters of P j lying over a subset S(W ) C Irr(W ) of very special characters of W.
This leads to a surprising cancellation between the number of characters of Pj over Irr(5"(W)) for different internal moduli. On the other hand, there is a bijection n atual that preserves the scales between Irr(G) and lrv (GUn ( q), 1) (where 1 denotes the central trivial character) which is limited to a bijection between the set of characters of the corresponding parabolics.
- General Facts
- Unitary groups
- The McKay Numbers
In summary, the set of irreducible characters of G comes in 1-1 correspondence with the set of pairs (s, A) where s is representative of a semi-simple class of G* and A is a unipotent character of (Cg*(S))* ; And the p-height of the character corresponding to (s, A) is equal to the p-height of A. The unipotent representations are studied by Lusztig and others in terms of the representations of the Weyl group. Similarly, considering GUn(q) as a subgroup of G L n(q2), the conjugation classes of elements in GUn{q) can also be described by elementary divisors, which turn out to be the powers of members in F.
Finally, if we denote the set of unipotent characters in G by Irru(G), then for finite reductive groups H i and d H 2 there is a natural 1 - 1. It is easy to check that each step in our construction of A is bijective, and so we obtain a 1-1 correspondence between Irr(C?) and the set of maps from F to P with desired properties.
Chapter 5 Evaluating Alternating Sums
- Overview
- The Combinatorial Approach
- Decomposable functions
This is a free abelian group B ( G ) with the set of equivalence classes of transitive G-sets as a basis, where two transitive G-sets G / H and G / K are equivalent if and only if H and K are conjugate in G. Direct a consequence of Lemma 5.1.2 is that the alternating sum of a G-stable function over a Brownian complex for G and some p is equal to the alternating sum over a Quillen or Bouc or Robinson complex, since all these complexes are G-homotopically equivalent. A subset X of elements in V is nondegenerate with respect to r if the restriction of r to X is not an identity mapping; in this case we say that r is nondegenerate on X.
In particular, a chain c 6 A("P) is nondegenerate with respect to r if it is nondegenerate as a subset of elements of V. Note that t r is an identity map if U is nondegenerate and r maps all elements in U if U is completely isotropic As required by Lemma 5.2.3, Cr arises on 0_1( P ) because QT(D) is obtained by inserting or deleting a suitable subgroup P into or from D and consequently P remains a finite term of QT(D).
Let P i be the subset of P on the set of subspaces U such that Rad(t7) > 0 together with 0, and P2 the subset of P i on the set of counting isotropic subspaces including 0. As required in Lemma 5.2 .3, Cr operates on 0-1(f7) since Ct(D) is obtained by inserting or deleting a totally isotropic subspace in or from D and therefore Cr(D) G 9~l (U). Let f be an H-stable function. where T consists of all the chains in V that contain a complement to W. Let Q be the subspace of V on the set of subspaces that is not the complete. Clearly, chains of up to and including isotropic subspaces are normal.
If c is nonsingular, then the nonsingularity of c is the dimension of the minimal nondegenerate member of c. Let T be the set of normal chains in A. Let An be the set of chains in V which do not contain a non-degenerate subspace. Observe that r consists of the n o n -s in g u la r normal chains of non-singular rank r in A. T) is the disjoint union of A r (resp.
- Tensor Modules
- Action on the Linear Modules
- Action on the Unitary Modules
Since Cp is identifiable by the additive group Fp, Hom(V, Cp) is G-isomorphic to Hompp (V, Fp). Check that cp is well defined and cp i-> cp defines a G-isomorphism between the two abelian groups. The fact that HompCVi F) is G-isomorphic to L(V i, V2, F) follows from the universal tensor product property.
Note that the proof of the first three G isomorphisms does not use the fact that t G is a product of two groups or V is a tensor module of G. It is easy to check that a t a: is a well-defined linear map between the two vector spaces. Remember also that a parabolic subgroup of a general linear group contains a Sylow p-subgroup of the general linear group.
On the other hand, recall that a t V f is the n atural modulus for Gr and P j T is the stabilizer in G + r of a flag of type J. By abuse of notation, we also say that r is labeled with c and w rite r = rc. where w is defined in th in the proposition. Moreover, let S^iV , J, 0) consist of the trivial character of V. Recall the definition of the singular c h a in s as well as the non-singular chains from Def.
By definition, there exists a 1-1 correspondence a v j from J, r) to the set A of Gr-orbits on nonsingular normal chains of nonsingular rank r7 in P (V n2) of type J.
- The Structure of the Unipotent Radicals
- Representations of Ui
- Action On the Central Modules
Moreover, Lj acts trivially on Z(Uj); Uj/Z(JJf) is the tensor module for Lj, described in the beginning of section 6.1. Indeed, all the statements in this lemma can be easily deduced from direct calculations, and we omit the proof. By laying a foundation for the natural module on which G acts, we can think of G as a matrix group.
Vq for L. Therefore, studying L-orbits on V ~ is equivalent to studying how fixed points with a in each orbit break up into V ~ orbits under L-action. If f H is a connected algebraic group over Fq and a is an endomorphism of f H, then the mapping g i-> cr{g)g~1 from H to itself is surjective. By Lemma 7.1.3, Lx is a semidirect product of connected algebraic groups and therefore connected.
I ^ m and let P = Pi be the maximal parabolic subgroup of G that stabilizes a 1-dimensional totally isotropic subspace in the natural modulus of G, U = Ui and P = L U , where L is a Levi complement of U . Set Z = Z{U). The linear representations of U are precisely those that contain Z in the kernel, and can therefore be identified with the representations of U /Z, which are examined in Sect. See for example (34.9) in [As], Explicitly, considered as a character of U, p is faithful and the unique irreducible character of U lies over r r.
According to lemma 7.1.3.3, K is contained in a maximal parabolic subgroup of G+1 that stabilizes a co-dimension r-subspace of V 1 . Let T be the set of chains in 'P containing a complement of w and Y(R) consists of the chains in V containing R. Let f be a G+1 stable function on P x X such that f x can be extended to a K stable function in the sense of Note 5.3.3. Also, by misuse of notation, we can say that r is labeled Ci in this case and write r = rCl.
Chapter 8 The Reduction Theorems
- General Discussion
- First Reduction
- Second Reduction
Recall from Section 2.3 that for C g C, S(dC) counts the number of distinct elements in d C that are considered a partition. Recall from Section 6.1 that each r G Irr(V) is assigned a rank which is invariant under the action of P (e ), and Irr(V, r) consists of the set of rank r characters in Irr(V' ). To simplify notation and to make it easier for readers to understand the proof, we prove (8.6) only for s = 1.
Indeed, C (J , m ax (J)) and C (/, £ ) are disjoint with each other since the minimal members of the convex sequences of different sets are different. But a paxabophic subgroup of a general unit contains a Sylow p-subgroup of the unit group, and the g-height of Gn = GUn(Fq) is. For s ^ 2, every P ( / ) is a product of s — 1 parabolic of general linear groups with a parabolic subgroup of a general unitary group.
Chapter 9 Completion of the Verification
- Further Reduction
- A Dimension Argument
- Completion of the Verification
In this section, we prove some technical results relating representations of certain subgroups from Gn to representations of certain subgroups of a unitary group of lower dimension. Again, a direct check with these two examples shows that 9 and £ are inverses of each other. The proof is analogous to the proof of Proposition 9.2.5, so the proof is omitted.
So later, when we prove Proposition 9.3.3 by induction on n, we can assume that the consequence is valid for all v. This splits the proof into two steps, namely the small case and the induction process.
Bibliography
