Remark. two routes from binary tetrahedral symmetry [fgap-001E]
✍️source

    The semidirect-product model and the quaternion model open two different routes from the same group. One rewrites the Group Algebra in interacting factor data. The other places the group inside \(\mathrm {SU}(2)\) and leads to the McKay graph.

    1. The Group-Algebra route

      Extend the conjugation action \(\alpha :C\to \operatorname {Aut}(Q)\) linearly to real-algebra automorphisms \[ \overline {\alpha }_c:\mathbb {R}[Q]\longrightarrow \mathbb {R}[Q]. \] On the vector space \(\mathbb {R}[Q]\otimes _{\mathbb {R}}\mathbb {R}[C]\), write a pure tensor as \(a\mathbin {\#}c\) and define \[ (a\mathbin {\#}c)(b\mathbin {\#}d) =a\,\overline {\alpha }_c(b)\mathbin {\#}cd, \] extending bilinearly. This convention defines the skew Group Algebra \(\mathbb {R}[Q]\rtimes _{\overline {\alpha }}C\); it is also a crossed product with trivial twisting cocycle.

      The basis map \[ [q]\mathbin {\#}c\longmapsto [qc] \] respects multiplication by the calculation in what the models of 2T retain, and unique factorization makes it a bijection. Hence \[ \mathbb {R}[Q]\rtimes _{\overline {\alpha }}C \mathbin {\cong _{\mathbb {R}\text {-alg}}}\mathbb {R}[T]. \] This is the Group-Algebra form of \(T\cong Q\rtimes _{\alpha }C\). The extension of group maps to Group-Algebra maps follows the convention in [sengupta2010representations, secs. 3.1--3.2, pp. 39--41].

      A module over this skew Group Algebra can be read as an \(\mathbb {R}[Q]\)-module \(M\) together with a linear action of \(C\) satisfying \[ c\mathbin {\cdot }(a\mathbin {\cdot }m) =\overline {\alpha }_c(a)\mathbin {\cdot } (c\mathbin {\cdot }m). \] The external model therefore exposes the compatibility needed to assemble representations from the two factors. It does not by itself decompose \(\mathbb {R}[T]\) into simple blocks. That requires further representation-theoretic input.

      2. The McKay route

        Under the standard matrix realization of the unit quaternions as \(\mathrm {SU}(2)\), the concrete group \(T\) becomes a finite subgroup of \(\mathrm {SU}(2)\). Let \(\tau _3\) be its faithful 2-dimensional complex representation. The McKay graph has one vertex for each irreducible complex representation \(\tau _i\); the number of edges from \(\tau _i\) to \(\tau _j\) is the multiplicity of \(\tau _j\) in \[ \tau _3\otimes \tau _i. \] For the binary tetrahedral group this graph is the affine Dynkin diagram \(\widetilde {E}_6\). The character table, the seven irreducible representations, and the tensor-product calculation are given in [stekolshchik2008notes, table A.12, prop. A.10, and ex. A.11, pp. 172--174].

        This graph connects the representation theory of \(2T\) with quivers and the ADE classification. The same source relates a binary polyhedral subgroup \(G\leq \mathrm {SU}(2)\) to the quotient \(\mathbb {C}^2/G\), its invariant algebra, and its Kleinian singularity in [stekolshchik2008notes, secs. A.3--A.4, pp. 158--161]. These are routes toward quotient geometry and related orbifold constructions. They are not consequences of the semidirect-product theorem alone.

        Neither route assigns physical meaning to \(Q\), \(C\), an algebra block, or a vertex of \(\widetilde {E}_6\). They supply mathematical structures on which a physical correspondence could be stated and tested. A proposed correspondence must still identify the action, representation, preserved structure, and physical interpretation.

        References

          Reference. Representations of Algebras and Finite Groups: An Introduction [sengupta2010representations]

          @unpublished{sengupta2010representations,
  author = {Sengupta, Ambar N.},
  title = {Representations of Algebras and Finite Groups: An Introduction},
  year = {2010},
  month = {May},
  note = {Course manuscript},
  url = {https://math.uchicago.edu/~margalit/repthy/sengupta\%2C\%20finitegrouprepresent.pdf}
}

          Reference. Notes on Coxeter Transformations and the McKay Correspondence [stekolshchik2008notes]

          @book{stekolshchik2008notes,
  author = {Stekolshchik, Rafael},
  title = {Notes on Coxeter Transformations and the McKay Correspondence},
  series = {Springer Monographs in Mathematics},
  publisher = {Springer Berlin, Heidelberg},
  year = {2008},
  doi = {10.1007/978-3-540-77399-3},
  isbn = {978-3-540-77398-6, 978-3-642-09604-4, 978-3-540-77399-3},
  url = {https://www-users.cse.umn.edu/~reiner/ReadingSeminars/McKaySeminar2016/Stekolshchik.pdf}
}

          Context

            Models and routes from binary tetrahedral structure [fgap-001F]
            📄PDF

            The concrete, internal, external, and abstract models of \(2T\) retain different data. Their comparison opens a Group-Algebra route through the semidirect factors and a representation-theoretic route through the McKay correspondence.

            Remark. what the models of 2T retain [fgap-001D]

              The same binary tetrahedral symmetry now has several models. They answer different questions.

              • The concrete model \(T\leq \mathbb {H}^{\times }\) retains quaternion coordinates, multiplication, conjugation, and norm. It is the model for explicit calculations.
              • The internal semidirect-product structure retains the same group \(T\), but also marks the subgroups \(Q,C\leq T\). The facts \[ T=QC,\qquad Q\mathrel {\trianglelefteq }T,\qquad Q\cap C=\{1\} \] say that every element of \(T\) has a unique factorization \(qc\). They expose how the two parts fit inside the concrete group.
              • The external model \(Q\rtimes _{\alpha }C\) replaces a quaternion by a pair \((q,c)\). It retains the factors and records their interaction in the conjugation action \(\alpha (c)(q)=cqc^{-1}\). Its multiplication is \[ (q,c)(q',c') =\bigl (q\alpha (c)(q'),cc'\bigr ). \]
              • An abstract model can specify the binary tetrahedral group without quaternion coordinates. This is convenient when only its group structure is needed, but coordinates and named factors must be restored by additional maps.

              The multiplication map \[ \mu :Q\rtimes _{\alpha }C\longrightarrow T,\qquad (q,c)\longmapsto qc \] is an isomorphism by the internal semidirect-product theorem. Indeed, \[ \mu \bigl ((q,c)(q',c')\bigr ) =q(cq'c^{-1})cc' =qcq'c' =\mu (q,c)\mu (q',c'). \] Thus the external multiplication is not an analogy: it is exactly the concrete multiplication written in factor coordinates. The two carriers are still different, so a theorem or construction passes between them only through an explicit isomorphism. If it refers to the named factors or their action, the transport must also record the corresponding compatibility.

              The internal and external constructions are developed in [fre2023discrete, sec. 4.2.13, pp. 62--63] and [isaev2018theory, sec. 1.4.2, pp. 58--61]. The concrete identification of \(2T\) with the Hurwitz unit group and \(Q_8\rtimes \mathbb {Z}/3\mathbb {Z}\) is given in [voight2021quaternion, sec. 11.2.4, p. 168].

              Remark. two routes from binary tetrahedral symmetry [fgap-001E]
              ✍️source

                The semidirect-product model and the quaternion model open two different routes from the same group. One rewrites the Group Algebra in interacting factor data. The other places the group inside \(\mathrm {SU}(2)\) and leads to the McKay graph.

                The Group-Algebra route

                  Extend the conjugation action \(\alpha :C\to \operatorname {Aut}(Q)\) linearly to real-algebra automorphisms \[ \overline {\alpha }_c:\mathbb {R}[Q]\longrightarrow \mathbb {R}[Q]. \] On the vector space \(\mathbb {R}[Q]\otimes _{\mathbb {R}}\mathbb {R}[C]\), write a pure tensor as \(a\mathbin {\#}c\) and define \[ (a\mathbin {\#}c)(b\mathbin {\#}d) =a\,\overline {\alpha }_c(b)\mathbin {\#}cd, \] extending bilinearly. This convention defines the skew Group Algebra \(\mathbb {R}[Q]\rtimes _{\overline {\alpha }}C\); it is also a crossed product with trivial twisting cocycle.

                  The basis map \[ [q]\mathbin {\#}c\longmapsto [qc] \] respects multiplication by the calculation in what the models of 2T retain, and unique factorization makes it a bijection. Hence \[ \mathbb {R}[Q]\rtimes _{\overline {\alpha }}C \mathbin {\cong _{\mathbb {R}\text {-alg}}}\mathbb {R}[T]. \] This is the Group-Algebra form of \(T\cong Q\rtimes _{\alpha }C\). The extension of group maps to Group-Algebra maps follows the convention in [sengupta2010representations, secs. 3.1--3.2, pp. 39--41].

                  A module over this skew Group Algebra can be read as an \(\mathbb {R}[Q]\)-module \(M\) together with a linear action of \(C\) satisfying \[ c\mathbin {\cdot }(a\mathbin {\cdot }m) =\overline {\alpha }_c(a)\mathbin {\cdot } (c\mathbin {\cdot }m). \] The external model therefore exposes the compatibility needed to assemble representations from the two factors. It does not by itself decompose \(\mathbb {R}[T]\) into simple blocks. That requires further representation-theoretic input.

                  The McKay route

                    Under the standard matrix realization of the unit quaternions as \(\mathrm {SU}(2)\), the concrete group \(T\) becomes a finite subgroup of \(\mathrm {SU}(2)\). Let \(\tau _3\) be its faithful 2-dimensional complex representation. The McKay graph has one vertex for each irreducible complex representation \(\tau _i\); the number of edges from \(\tau _i\) to \(\tau _j\) is the multiplicity of \(\tau _j\) in \[ \tau _3\otimes \tau _i. \] For the binary tetrahedral group this graph is the affine Dynkin diagram \(\widetilde {E}_6\). The character table, the seven irreducible representations, and the tensor-product calculation are given in [stekolshchik2008notes, table A.12, prop. A.10, and ex. A.11, pp. 172--174].

                    This graph connects the representation theory of \(2T\) with quivers and the ADE classification. The same source relates a binary polyhedral subgroup \(G\leq \mathrm {SU}(2)\) to the quotient \(\mathbb {C}^2/G\), its invariant algebra, and its Kleinian singularity in [stekolshchik2008notes, secs. A.3--A.4, pp. 158--161]. These are routes toward quotient geometry and related orbifold constructions. They are not consequences of the semidirect-product theorem alone.

                    Neither route assigns physical meaning to \(Q\), \(C\), an algebra block, or a vertex of \(\widetilde {E}_6\). They supply mathematical structures on which a physical correspondence could be stated and tested. A proposed correspondence must still identify the action, representation, preserved structure, and physical interpretation.

                    Backlinks

                      Theorem. Clifford homogeneity and the semidirect route [fgap-001Q]

                        Let \(N\trianglelefteq G\) and let \(V\) be an irreducible complex representation of \(G\). If \(W_1,\ldots ,W_r\) represent the distinct conjugates of an irreducible constituent of \(V|_N\), then for 1 positive integer \(e\), \[ V|_N\cong e(W_1\oplus \cdots \oplus W_r). \] Thus restriction is homogeneous along a single orbit. The common multiplicity is an additional assertion beyond the single orbit in Clifford restriction. This is Clifford's theorem, [james2001representations, thm. 20.8, pp. 216--217].

                        The inertia subgroup is the intermediate group where one studies whether \(W\) extends, possibly projectively, before inducing to \(G\). For semidirect products, this reduces the problem to the action of the complement. For \(Q_8\rtimes C_3\cong 2T\), it connects the normal quaternion subgroup to the skew-Group-Algebra model in two routes from binary tetrahedral symmetry. This theorem does not supply the full character table of \(2T\).

                        Example. semidirect products make Clifford theory concrete [fgap-0020]

                          Suppose \(G=N\rtimes H\). Conjugation by \(H\) permutes the irreducible constituents of \(V|_N\). Their stabilizers are the inertia subgroups, so the orbit and homogeneity statements in the single orbit in Clifford restriction and Clifford homogeneity and the semidirect route reduce the search for irreducible \(G\)-representations to stabilizer data, projective extension, and induction. Clifford theory thereby gives a character-theoretic route for semidirect products.

                          For \(Q_8\rtimes C_3\cong 2T\), the action and ordinary skew-Group-Algebra realization are recorded in the central split of the real Group Algebra of 2T and two routes from binary tetrahedral symmetry. They supply the normal-subgroup and algebraic input to this route, but do not give the full character table of \(2T\).

                          Remark. the inertia obstruction is projective [fgap-001T]

                            For a constituent \(W\) with inertia subgroup \(I=I_G(W)\), choose intertwiners \(T_i:W\to {}^iW\). Their composites need not satisfy the group law strictly. Instead one obtains scalars \[ T_iT_j=\alpha (i,j)T_{ij}. \] Associativity makes \(\alpha \) a factor set. Changing the intertwiners changes \(\alpha \) by a coboundary. The resulting cohomology class measures the obstruction to replacing the projective action by an ordinary action.

                            Twisted Group Algebras therefore enter the Clifford-theory route. The ordinary skew Group Algebra in two routes from binary tetrahedral symmetry corresponds to trivial twisting; it should not be identified with a general \(k_\alpha [G]\). The extension and induction analysis is developed in [lux2010representations, secs. 3.6--3.7, pp. 222--240].

                            Related

                              Remark. what the models of 2T retain [fgap-001D]

                                The same binary tetrahedral symmetry now has several models. They answer different questions.

                                • The concrete model \(T\leq \mathbb {H}^{\times }\) retains quaternion coordinates, multiplication, conjugation, and norm. It is the model for explicit calculations.
                                • The internal semidirect-product structure retains the same group \(T\), but also marks the subgroups \(Q,C\leq T\). The facts \[ T=QC,\qquad Q\mathrel {\trianglelefteq }T,\qquad Q\cap C=\{1\} \] say that every element of \(T\) has a unique factorization \(qc\). They expose how the two parts fit inside the concrete group.
                                • The external model \(Q\rtimes _{\alpha }C\) replaces a quaternion by a pair \((q,c)\). It retains the factors and records their interaction in the conjugation action \(\alpha (c)(q)=cqc^{-1}\). Its multiplication is \[ (q,c)(q',c') =\bigl (q\alpha (c)(q'),cc'\bigr ). \]
                                • An abstract model can specify the binary tetrahedral group without quaternion coordinates. This is convenient when only its group structure is needed, but coordinates and named factors must be restored by additional maps.

                                The multiplication map \[ \mu :Q\rtimes _{\alpha }C\longrightarrow T,\qquad (q,c)\longmapsto qc \] is an isomorphism by the internal semidirect-product theorem. Indeed, \[ \mu \bigl ((q,c)(q',c')\bigr ) =q(cq'c^{-1})cc' =qcq'c' =\mu (q,c)\mu (q',c'). \] Thus the external multiplication is not an analogy: it is exactly the concrete multiplication written in factor coordinates. The two carriers are still different, so a theorem or construction passes between them only through an explicit isomorphism. If it refers to the named factors or their action, the transport must also record the corresponding compatibility.

                                The internal and external constructions are developed in [fre2023discrete, sec. 4.2.13, pp. 62--63] and [isaev2018theory, sec. 1.4.2, pp. 58--61]. The concrete identification of \(2T\) with the Hurwitz unit group and \(Q_8\rtimes \mathbb {Z}/3\mathbb {Z}\) is given in [voight2021quaternion, sec. 11.2.4, p. 168].