The above exceptional isomorphisms involving the projective special linear groups are almost all of the exceptional isomorphisms between families of finite simple groups; the only other exceptional isomorphism is PSU ( 4, 2 ) C " PSp ( 4, 3 ), between a projective special unitary group and a projective symplectic group.
32.
Young diagrams also parametrize the irreducible polynomial representations of the general linear group ( when they have at most nonempty rows ), or the irreducible representations of the special linear group ( when they have at most nonempty rows ), or the irreducible complex representations of the special unitary group ( again when they have at most nonempty rows ).
33.
For instance, the special orthogonal group SO ( 3 ) and the special unitary group SU ( 2 ) give rise to the same Lie algebra, which is isomorphic to "'R "'3 with the cross-product, while SU ( 2 ) is a simply-connected twofold cover of SO ( 3 ).
34.
For example, P?U is used for the semilinear analogs of the projective special unitary group PSU . Note however, that it is only recently noticed that these generalized semilinear groups are not well-defined, as pointed out in isomorphic classical groups " G " and " H " ( subgroups of SL ) may have non-isomorphic semilinear extensions.
35.
There is a division algebra " D " with center " l " and degree over " l " 3 or 1, with an involution of the second kind which restricts to the nontrivial automorphism of " l " over " k ", and a nontrivial Hermitian form on a module over " D " of dimension 1 or 3 such that " G " is the special unitary group of this Hermitian form . ( As a consequence of and the work of Cartwright and Steger, " D " has degree 3 over " l " and the module has dimension 1 over " D " . ) There is one real place of " k " such that the points of " G " form a copy of PU ( 2, 1 ), and over all other real places of " k " they form the compact group PU ( 3 ).
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