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A SHORT NOTE ON INDUCED REPRESENTATIONS FULVIO GESMUNDO This note is result of discussions with friends and colleagues, in particular Eugenio Giannelli e Christian Ikenmeyer. It assume basic knowledge in representation theory of finite groups. Some useful reference at the end. Let G be a finite group and let H be a subgroup (in general not normal) with G k = |G : H|. Let W be a left representation of H and denote by (W ) ↑H the induced representation of G. In this note we give an explicit construction of G (W ) ↑H and we show that, as a G-modules, it only depends on the conjugacy class of H as a subgroup of G. Construction of the induced representation. Let T = {g1 , . . . , gk } be a set of representatives for the left cosets of H in G, namely g1 H, . . . , gk H are the cosets of H in G. Write g = (g1 , . . . , gk ). Let Mg = hg1 , . . . , gk i be a vector space having T as basis. We define functions h(·) and j(·) defined as follows: for g in G, write h(g) ∈ H and j(g) = 1, . . . , k for the unique elements such that g = gj(g) h(g). G Then define (W ) ↑H as Mg ⊗ W with the left G-action given as follows on basis elements and extended by linearity: g · (gi ⊗ w) = gj(ggi ) ⊗ h(ggi )w. The following calculation shows that the left action is well defined. Let g 0 , g ∈ G. Then: (g 0 g) · (gi ⊗ w) = (gj(g0 ggi ) ⊗ h(g 0 ggi )w; g 0 · [g · (gi ⊗ w)] = g 0 · [gj(ggi ) ⊗ h(ggi )w] = = gj(g0 gj(gg ) ) ⊗ h(g 0 gj(ggi ) )h(ggi )w i j(g 0 gj(ggi ) ) j(g 0 ggi ) and the relations = and h(g 0 gj(ggi ) )h(ggi ) = h(g 0 ggi ) follow by uniqueness of the expression of g 0 ggi as element of a coset of H with representatives g (namely by the good definition of j(·) and h(·)); indeed g 0 ggi = gj(g0 ggi ) h(g 0 ggi ) g 0 ggi = gj(g0 gj(gg ) ) h(g 0 gj(ggi ) )h(ggi ). i 1 2 FULVIO GESMUNDO Notice that changing set of representatives provides an isomorphic representation. Let g0 = (g10 , . . . , gk0 ), with gi H = gj0 H. The two realizations Mg ⊗ W G and Mg0 ⊗ W of (W ) ↑H are isomorphic via the map defines as follows: let hi ∈ H such that gi = gi0 ki ; then define ϕ : M g ⊗ W → Mg 0 ⊗ W gi ⊗ w 7→ gi0 ⊗ ki w. The map ϕ is a G-equivariant isomorphism. Write j 0 (·) and h0 (·) for functions analogous to j(·) and h(·), calculated with respect to g0 . The following calculation checks equivariancy: 0 ϕ(g · (gi ⊗ w)) = ϕ(gj(ggi ) ⊗ h(ggi )w) = gj(gg ⊗ kj(ggi ) h(ggi )w; i) g · ϕ(gi ⊗ w) = g · (gi0 ⊗ ki w) = gj0 0 (gg0 ) ⊗ h0 (ggi0 )ki w. i j 0 (ggi0 ) The two relations = j(ggi ) and kj(ggi ) h(ggi ) = h0 (ggi0 )ki follow from the uniqueness of the expression of the element ggi in a coset of H with representatives g; indeed we have ggi = ggi0 ki = gj0 0 (gg0 ) h0 (ggi0 )ki , i 0 ggi = gj(ggi ) h(ggi ) = gj(gg k h(ggi ), i ) j(ggi ) 0 so gj0 0 (gg0 ) = gj(gg and h0 (ggi0 )ki = kj(ggi ) h(ggi ). i) i e is a conjugate of H in G, Conjugate subgroups. Now suppose that H −1 e = τ Hτ or equivalently Hτ = τ H. e namely there exists τ ∈ G such that H The isomorphism e →H ψτ : H e h 7→ τ e hτ −1 e via e f makes W into a left module for H h · w = ψτ (e h) · w. We denote by W this representation. f ) ↑G ' (W ) ↑G as G-representations. We want to show that (W H e H e = (e Define g g1 , . . . , gek ) to be the set of representatives for the left cosets of e e define a set of H defined by gei = gi τ . It is immediate that the elements of g e = gej H, e if and only if gi τ H e = gj τ H e if and only representatives: indeed gei H if gi Hτ = gj Hτ if and only if gi H = gj H if and only if i = j. Write e h(·) and e j(·) for functions analogous to h(·) and j(·) relative to cosets e of H with representatives g. f. f ) ↑G = Mge ⊗ W We have a realization (W e H A SHORT NOTE ON INDUCED REPRESENTATIONS 3 Define f → Mg ⊗ W T : Mge ⊗ W gei ⊗ w 7→ gi ⊗ w. The map T defines an isomorphism of vector spaces. We show that it is G-equivariant. Let g ∈ G. We have T (g · (e gi ⊗ w)) = T (e ge ⊗e h(ge gi )w) = j(ge gi ) = T (e gej(gegi ) ⊗ ϕτ (e h(ge gi ))w) = gej(gegi ) ⊗ τ e h(ge gi )τ −1 w, g · T (e gi ⊗ w) = g · (gi ⊗ w) = gj(ggi ) ⊗ h(ggi )w; to prove equivariancy, we need to show j(ggi ) = e j(ge gi ) h(ggi ) = τ e h(ge gi )τ −1 . We consider the element ge gi τ −1 . We have ge gi τ −1 = ggi τ τ −1 = ggi , ge gi τ −1 = geej(gegi )e h(ge gi )τ −1 = gej(gegi ) τ e h(ge gi )τ −1 , and this leads to the conclusion we are seeking as τ e h(ge gi )τ −1 is an element G G f ) ↑ ' (W ) ↑ as G-representations. of H. This shows that (W H e H References [A] J.L. Alperin Local Representation Theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, 1986. [FH] W. Fulton and J. Harris Representation theory: a first course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. [JK] G.D. James, A. Kerber The Representation Theory of the Symmetric Group, Addison-Wesley, London, 1980. [R] M. Reeder Notes on Representation Theory, Lecture Notes available at https://www2.bc.edu/˜reederma/RepThy.pdf, 2014. E-mail address: [email protected] Department of Mathematics - Texas A&M University - College Station - TX - 77840