Exceptional object

s, seen here in an illustration from Johannes Kepler's Mysterium Cosmographicum (1596), are an early example of exceptional objects. The symmetries of three-dimensional space can be classified into two infinite families—the cyclic and dihedral symmetries of n-sided polygons—and five exceptional types of symmetry, namely the symmetry groups of the Platonic solids. In mathematics, an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series; these are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups.

s, seen here in an illustration from Johannes Kepler's Mysterium Cosmographicum (1596), are an early example of exceptional objects. The symmetries of three-dimensional space can be classified into two infinite families—the cyclic and dihedral symmetries of n-sided polygons—and five exceptional types of symmetry, namely the symmetry groups of the Platonic solids.]] In mathematics, an exceptional object is one of finitely many exceptions to some classification of objects. Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series; these are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others. A related phenomenon is exceptional isomorphism, when two series are in general different, but agree for some small values. For example, spin groups in low dimensions are isomorphic to other classical Lie groups. Among families of finite simple groups, the only example is in the automorphisms of the symmetric and alternating groups: for n \geq 3, n \neq 6 the alternating group A_n has one outer automorphism (corresponding to conjugation by an odd element of S_n ) and the symmetric group S_n has no outer automorphisms. However, for n=6, there is an exceptional outer automorphism of S_6 (of order 2), and correspondingly, the outer automorphism group of A_6 is not C_2 (the group of order 2), but rather C_2 \times C_2 , the Klein four-group. If one instead considers A_6 as the (isomorphic) projective special linear group \operatorname{PSL}(2,9) , then the outer automorphism is not exceptional; thus the exceptional-ness can be seen as due to the exceptional isomorphism A_6 \cong \operatorname{PSL}(2,9). This exceptional outer automorphism is realized inside of the Mathieu group M_{12} and similarly, M_{12} acts on a set of 12 elements in 2 different ways. Among Lie groups, the spin group \operatorname{Spin}(8) has an exceptionally large outer automorphism group (namely S_3 ), which corresponds to the exceptional symmetries of the Dynkin diagram D_4 . This phenomenon is referred to as triality. The exceptional symmetry of the D_4 diagram also gives rise to the Steinberg groups. Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to 8 and 24 dimensions, noting that 24 = 8 · 3. By contrast, the pariah groups stand apart, as the name suggests. Triality occurs for Spin(8), which also connects to 8 · 3  = 24. Likewise, exceptional objects related to the number 24 include the following. The Leech lattice is 24-dimensional. Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster. The exceptional Jordan algebra has a representation in terms of 24×24 real matrices together with the Jordan product rule. These objects are connected to various other phenomena in math which may be considered surprising but not themselves "exceptional". For example, in algebraic topology, 8-fold real Bott periodicity can be seen as coming from the octonions. In the theory of modular forms, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions to the Monster group.

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