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19 Mellis, O., G6teborgs Kungl. Vetenkaps. och Vitterhets-Samhdlles Handl., 6, Folgden, Ser. B, Bd. 5. No. 13,45 (1948). 20 Zeigler, J. M., W. D. Athearn, and H. Small, Deep Sea Res., 4, 238-249 (1957). 21 Rex, Robert W., and Edward D. Goldberg, Tellus, 10, 153 (1958). 22 Buddhue, John Davis, Univ. of New Mex. Publications in Meteoritics, No. 2, 1 (1950). 23 Banfield, A. F., C. H. Behre, Jr., and David St. Clair, Bull. Geol. Soc. of Am., 67, 215 (1956). 24 Lewis, G. E., H. J. Tchopp, and J. G. Marks, Geol. Soc. Am. Mem., 65, 251 (1956). 25 Cromwell, T., R. B. Montgomery, and E. D. Stroup, Science, 117, 648 (1954). 26i Fisher, R. L., I.G. Y. General Report Series, 2, 58 (1958). 27 Ericson, D. B., and G. Wollin, Deep Sea Res., 3, 104 (1956). 28 Smith, George I., and W. P. Pratt, Geol. Surv. Bull., 104fA, 1-62, 1957. 29 Urey, H. C., Nature, 179, 556-557 (1957).
ON THE NUMBER OF IRREDUCIBLE CHARACTERS OF FINITE GROUPS IN A GIVEN BLOCK BY RICHARD BRAUER AND WALTER FEIT
HARVARD UNIVERSITY AND CORNELL UNIVERSITY Communicated January 16, 1959 Let @ be a group of finite order g and let p be a fixed prime number. The blocks of irreducible characters of @5 with regard to p and their significance for the arith- metic in the group ring of @ have been discussed in several previous publications.' In particular, it has been stated that the number m of ordinary irreducible char- acters in a p-block of defect d is at most equal to pd(d+1)/2. In the present note, this result will be improved. We shall show: THEOREM 1. Let B be a p-block of defect d of a finite group @. The number m of ordinary irreducible characters x1, X22 ... Xm in B satisfies the inequality m < /4p2d + 1. (1) Our new proof which we shall now sketch is far more elementary than the pre- vious proof of the weaker result. Let B consist of the modular irreducible char- acters pj, 4p2, . ...p,,. For p-regular elements R of @5, i.e., for elements R of @ whose order is prime to p, we have formulas n xi(R = E djj1p(R), (1 < i < m) (2) j7=1 where the dij are nonnegative rational integers, the decomposition numbers of @5 belonging to B. Set aij = (pd/g) a xi(R)xj(R), (1 < i, j < m) (3) R where R ranges over all p-regular elements of @. Let A denote the m X m matrix (aij) and let D denote the m X n matrix (dij). If D' denotes the transpose of D, then C = D'D is the n X n matrix of Cartan invariants of @ belonging to B. It follows from (2), (3), and the orthogonality relations for the modular group char- acters that A = pdDC-'D'. (4) Downloaded by guest on September 29, 2021 362 MATHEMATICS: BRAUER AND FEIT PROC. N. A. S.
It is known that C has integral coefficients and that its largest elementary divisor is pd. Now (4) shows that A has integral coefficients. Clearly, A is a symmetric matrix. The nth determinantal divisor of the m X n matrix D is 1.2 Using the normal form of D, we deduce from (4) that there exists an m X m matrix U with integral coefficients and determinant 1 such that
UAU' = Pd( 0) 0/ It follows that not all coefficients of UA U' are divisible by p and hence not all coefficients of A are divisible by p. Let IR1, R2 . .. , Rh4 be a system of representatives for the classes of p-regular conjugate elements of 5. If the class of Ra consists of g9 elements, and if xi = xi (1) is the degree of xi, then it is well known that the number wi(Ra) = gaXi(Ra)/xi is an algebraic integer. Introducing these quantities in (3), we obtain h aij = (pd/g)X Ej i(Ra);j(Ra). (5) If Xr is another irreducible character in B and if Cwr has the analogous significance as wi, then modulo a suitable prime ideal divisor p of p, we have wi(Ra) = Wr(R.) (mod p). Now, (5) yields easily the congruence (g/pd) (aijixi) = (g/pd) (aTj/xT) (mod p). Let v denote the p-adic exponential valuation, v(p) = 1. By the definition of the defect d of a block B, we can set v(xi) = v(g)-d +Xi (6) with Xf > 0 for 1 < i < m. We shall term the integer Xi the height of the char- acter Xi in B. There exist characters of height 0 in B. We choose Xr as such a character, Xr = 0. If congruences are taken as congruences in the ring of local integers for p, our result becomes aft (xi/xr)ari (mod pl+Xi). Using the symmetry of aij, we find a1= (xixjl/xr2)arr (mod pl+xi). (7) It follows that arr is not divisible by p, since otherwise all aij would be divisible by p, and we have already seen that this is not true. Takingj = r in (7), we obtain v(air) = Xi. (8) In particular, air $ 0 for 1 < i < m. Downloaded by guest on September 29, 2021 VOL. 45, 1959 MATHEMATICS: BRA UER AND FEIT 363
It follows from (4) together with D'D = C that A2 = pdA. (9) Hence m E air2 = pdarr. (10) i= 1 Since air $ 0, this yields
(m - 1) + a,.2, pdarr. The maximum of pdX-x2 is obtained for x = pd/2 and hence m- 1
Then as shown by (8), mx terms air in (10) contain p with the exact exponent pa. Hence the method used above yields (MO-1) + m1p2 + m2p4 + < 1/4 p2d (13) In particular m?, < 1/4p2d - 2 (14) for X > 0, as it is easy to see that mO > 1. On the other hand, if B contains characters Xi of positive height Xi = X > 0, then mi terms aji in (11) contain p with the exact exponent X. Hence mo . 1/4pd - ^ (15) On combining (12), (13), and (15) we obtain THEOREM 3. If the p-block B of defect d contains irreducible characters Xi of posi- tive height, the number m of irreducible characters in B is less than l/2p2d 2- Downloaded by guest on September 29, 2021 364 MATHEMATICS: BRA UER AND FEIT PROC. N. A. S.
So far we have only used p-regular elements R of @. If P now is an element of @ whose order is a power pa . 1 of p, and if R is a p-regular element of the central- izer Q,;(P) of P, the formula (2) can be generalized to Xi(PR) = E di! sejp(R), J where the f] are the modular irreducible characters of ((P) and where the df! are algebraic integers of the field 1,a of the path roots of unity. If we consider only xi belonging to a fixed p-block B of @, only characters P have to be taken which belong to a well-determined set of p-blocks B of ((P). We set ail = (pd/n(P)) EZRxi(PR)Xij(PR), where n(P) is the order of Q(P) and where R ranges over the p-regular elements of S(P). For P = 1, we have afP = aij. Let AP be the matrix of the af!, DP the matrix of the di, and 'P the direct sum of the matrices C of the Cartan invariants of the blocks B of E(P) associated with B. Then (4) can be generalized to
AP = pdDP(CP)-1(DP)t It follows that the aP are algebraic integers of the field Q2a. Similarly to (9), we have (AP)2 = pdAP. There is also a form in which (7) can be generalized. If P ranges over a system of representatives for the classes of conjugate elements whose order is a power of p, we have Eai-j = pdbj. Finally, there exists a simple connection between the p-blocks of C(P) and those of &(P)/{P}. We now state a number of results without proof which can be shown by means of these formulas. If we use known results4 on blocks of defect 0 and 1 and Theo- rem 3, we obtain the following improvement of Theorem 1. THEOREM 1*. If B, p, d, m have the same significance as in Theorem 1, then m < pdfor d =O. 1, 2 and m < p2d-2 for d > 2. d Probably, the inequality m < pd holds for all d, but we have not been able to prove this. However, for certain types of defect groups S of the block B, our re- sults can be improved. For instance, we can prove: THEOREM 4. Ifthe defect group Z of the block B in Theorem 1 is cyclic, then m < pd. Actually, there is a connection between the structure of the defect group ) of the block B and the maximal height X of characters in B. THEOREM 5. Let Z of order pd be the defect group of the p-block B of @5 and assume that there exist u elements in the center of Z such that no two of them are conjugate in @5. If B contains characters x ofheight X > 0, then U < pd-. (16) If p = 2, then (16) can be replaced by u < 2d 1 (16*) COROLLARY. If B is a block offull defect d = v(g), i.e., if the defect group Z is a Downloaded by guest on September 29, 2021 VOL. 45, 1959 MATHEMATICS: F. E. BROWDER 365
p-Sylow group of @ and if the center of Z contains an element P of order pa, then B can contain characters ofheight X only if X
ESTIMATES AND EXISTENCE THEOREMS FOR ELLIPTIC BOUNDARY VALUE PROBLEMS* BY FELIX E. BROWDER
DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY Communicated by Einar Hille, January 12, 1959 Let A be a linear elliptic differential operator of order 2m (m > 1) with complex- valued coefficients defined on an open subset G of Euclidean n-space En. It is our purpose in the present note to discuss the existence and regularity of solutions of a general family of boundary-value problems of local type for A on G. An impor- tant aspect of our discussion is that it assumes no regularity conditions on the coefficients of A except continuity, makes no semiboundedness assumption on the boundary-value problem, and yields information on the nature of A realized as an operator in LP(G) under null boundary conditions for unbounded as well as bounded domains G. The key tool of our discussion is a family of a priori estimates for the derivatives of solutions in LP(G) (as well as in Holder norm, provided the coeffi- cients are Holder continuous). Since the existence results are obtained under extremely weak assumptions upon the regularity of the coefficients of A, they are a useful step toward the construction of a theory of linear equations sufficient for the attack upon boundary value problems for nonlinear elliptic equations. The results which are discussed below are immediately extendable to the general class of elliptic systems under analogous hypotheses. In a note to follow, we shall dis- cuss the stronger conclusions that may be obtained under similar hypotheses for the Dirichlet problem for strongly elliptic equations and systems (as well as other semi- bounded problems) and derive from these, existence theorems for mixed initial- boundary value problems for temporally inhomogeneous equations of propagation. We shall use the notation Dj = i-1 a/axj, while for an n-tuple a of nonnegative (al, . . . , an), we set De' = Dial ... D:R, a = Zjaj. Similarly, for a complex n-vector r = (id, ... ., n), ra = W" . . Rnn.. In terms of this notation, the differ- ential operator A may be written as E la <2m aa(x)Da, and its characteristic form is, a(x, I) = E la I=2maa(x).a If B1, ... X Bm are m differential operators of order <2m defined on a neighborhood of the boundary of G, each Bj may be written as Downloaded by guest on September 29, 2021