Example. the central split of the real Group Algebra of 2T [fgap-0015]
Example. the central split of the real Group Algebra of 2T [fgap-0015]
Let \(B\) be the abstract binary tetrahedral group, let \(z\in B\) be its distinguished central involution, and put \[ A=\mathbb {R}[B]. \] The group element \(z\), its basis image \([z]\in A\), and the scalar \(-1_A\) are different kinds of objects. The image \([z]\) is central in \(A\) and satisfies \([z]^2=1_A\), by a central group element inside a Group Algebra.
The two elements \[ e_+=\frac {1_A+[z]}2,\qquad e_-=\frac {1_A-[z]}2 \] therefore satisfy \[ e_+^2=e_+,\qquad e_-^2=e_-,\qquad e_+e_-=e_-e_+=0,\qquad e_++e_-=1_A. \] They are central by the two idempotents of a central involution. Hence left multiplication by \(e_+\) and \(e_-\) gives two complementary projections on the regular \(A\)-module. This specializes the generic central-involution split to \(\mathbb {R}[B]\); no representation-theoretic classification is used.
The Group Algebra convention is developed in [sengupta2010representations, secs. 3.1--3.3, pp. 39--42]. The projection interpretation of complementary idempotents is compared with [webb2007finite, exercise 2.7, p. 15].