[過去ﾛｸﾞ] ガロア第一論文及びその関連の資料スレ (1002ﾚｽ)

ガロア第一論文及びその関連の資料スレ http://rio2016.5ch.net/test/read.cgi/math/1615510393/

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350: １３２人目の素数さん [] 2023/02/11(土) 23:42:17.80 ID:cDdl8Z4s >>349 つづき Now both theories have gamed considerable interest in their respective fields, mathematics and physics. Despite the fact that these theories have no common motivations, goals or problems, their formal similarity goes remarkably far. In this paper we discuss primarily the mathematical theory. For a review of the physical theory see the paper of J. Schwarz in this volume [27] Then in [9, 28] the "vertex construction" was found for the whole class of affine Lie algebras and the similarity became a precise correspondence. つづく http://rio2016.5ch.net/test/read.cgi/math/1615510393/350

351: １３２人目の素数さん [] 2023/02/11(土) 23:42:38.02 ID:cDdl8Z4s >>350 つづき Let us fix a light-cone element c ∈ Δ such that there are no real roots orthogonal to it. Such a vector exists and the set L = [a ∈ ΔR: (a, c) =1} is isomorphic to the unique even unimodular lattice of rank 24, which does not contain elements of length √2 [2]. We denote by V1,c, the space Σα∈L V1,c,α. Then the character of V1,c, is j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21) It was noticed by McKay that the number 196884 exceeds by only one the dimension of the minunal representation of F1. Conway and Norton [3] conjectured that there is a natural graded representation of F] with the character (4.21) minus 24. First Garland [12] and Kac [17] independently tried to construct i7i in a space isomorphic to ^ . The first problem was to obtain a representation of one important subgroup C=2^+l ' ・(・0)/±1, where -0 is the automorphism group of the Leech lattice. It is easy to construct another group C' = 224 ・ (-0) (= (2M+1/±1) ・ (-0)). Using one observation of Griess, Kac [18] succeeded in passing from C' to C. The last question is: Where is the whole group F\1 Recently, important progress has been made in answer to this question [10]. Turning again to the dual resonance models gives a hint as to the answer. Physicists know that m the contmuous version of F; ^ the obvious action of the group 0(24) can be extended to the bigger group 0(25). This extension becomes apparent only if we return to the bigger space V1+. Whether this unusual phenomenon corresponds to the extension of C to F1 will become clear in the future. (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1615510393/351

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