Representation theory and Gelfand pairs Dmitry Gourevitch FUN seminar, WIS http://www.wisdom.weizmann.ac.il/~dimagur/ December 2013 Gourevitch Representation theory and Gelfand pairs December 2013 1 / 7 Definition A representation p of a group G on a (complex) vector space V is an assignment to every element g 2 G of an invertible linear operator p(g) such that p(gh) is the composition of p(g) and p(h). Example An action of G on a set X defines a representation on the space C[X ] of functions on X by (p(g)f )(x) := f (g −1x). Representation theory Advanced Linear Algebra Linear Algebra in presence of symmetries Gourevitch Representation theory and Gelfand pairs December 2013 2 / 7 Example An action of G on a set X defines a representation on the space C[X ] of functions on X by (p(g)f )(x) := f (g −1x). Representation theory Advanced Linear Algebra Linear Algebra in presence of symmetries Definition A representation p of a group G on a (complex) vector space V is an assignment to every element g 2 G of an invertible linear operator p(g) such that p(gh) is the composition of p(g) and p(h). Gourevitch Representation theory and Gelfand pairs December 2013 2 / 7 Representation theory Advanced Linear Algebra Linear Algebra in presence of symmetries Definition A representation p of a group G on a (complex) vector space V is an assignment to every element g 2 G of an invertible linear operator p(g) such that p(gh) is the composition of p(g) and p(h). Example An action of G on a set X defines a representation on the space C[X ] of functions on X by (p(g)f )(x) := f (g −1x). Gourevitch Representation theory and Gelfand pairs December 2013 2 / 7 The same holds for the compact group S1. The basis vectors are fk (q) = exp(ikq). The decomposition of a function with respect to this basis is called Fourier series. For the group SO(3) of rotations in the space this does not hold, neither for C[SO(3)] nor for C[S2] (functions on the sphere) One-dimensional representations and Fourier series For the cyclic finite group Z/nZ, the space C[G ] has a basis consisting of joint eigenvectors for the whole representation. The basis vectors are fk (m) = exp(2pikm/n). The decomposition of a function with respect to this basis is called discrete Fourier transform. Gourevitch Representation theory and Gelfand pairs December 2013 3 / 7 For the group SO(3) of rotations in the space this does not hold, neither for C[SO(3)] nor for C[S2] (functions on the sphere) One-dimensional representations and Fourier series For the cyclic finite group Z/nZ, the space C[G ] has a basis consisting of joint eigenvectors for the whole representation. The basis vectors are fk (m) = exp(2pikm/n). The decomposition of a function with respect to this basis is called discrete Fourier transform. The same holds for the compact group S1. The basis vectors are fk (q) = exp(ikq). The decomposition of a function with respect to this basis is called Fourier series. Gourevitch Representation theory and Gelfand pairs December 2013 3 / 7 One-dimensional representations and Fourier series For the cyclic finite group Z/nZ, the space C[G ] has a basis consisting of joint eigenvectors for the whole representation. The basis vectors are fk (m) = exp(2pikm/n). The decomposition of a function with respect to this basis is called discrete Fourier transform. The same holds for the compact group S1. The basis vectors are fk (q) = exp(ikq). The decomposition of a function with respect to this basis is called Fourier series. For the group SO(3) of rotations in the space this does not hold, neither for C[SO(3)] nor for C[S2] (functions on the sphere) Gourevitch Representation theory and Gelfand pairs December 2013 3 / 7 Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Proof. Definition A representation is called irreducible if the space does not have invariant subspaces. Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Lemma (Schur) Proof. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Proof. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Any non-zero morphism of irreducible representations is invertible. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Proof. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Proof. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Proof. Ker T , Im T are subrepresentations. Definition A representation is called irreducible if the space does not have invariant subspaces. Definition A morphism between representations (p, V ) and (t, W ) of a group G is a linear operator T : V ! W s. t. T ◦ p(g) = t(g) ◦ T for any g 2 G. Lemma (Schur) Any non-zero morphism of irreducible representations is invertible. Any morphism of an irreducible finite-dimensional representation into itself is scalar. Proof. Ker T , Im T are subrepresentations. T has an eigenvalue l, thus T − l Id is not invertible, thus T − l Id = 0. Hn is an irreducible representation of SO(3) 2 2 L¥ L (S ) = cn=0Hn, Every irreducible representation of SO(3) is isomorphic to Hn for some n. Theorem Spherical harmonics Hn :=the space of homogeneous harmonic polynomials of degree n in three variables. Harmonic means that they vanish under the Laplace ¶2 ¶2 ¶2 operator D = 2 + 2 + 2 . ¶x1 ¶x2 ¶x3 2 2 L¥ L (S ) = cn=0Hn, Every irreducible representation of SO(3) is isomorphic to Hn for some n. Spherical harmonics Hn :=the space of homogeneous harmonic polynomials of degree n in three variables. Harmonic means that they vanish under the Laplace ¶2 ¶2 ¶2 operator D = 2 + 2 + 2 . ¶x1 ¶x2 ¶x3 Theorem Hn is an irreducible representation of SO(3) Every irreducible representation of SO(3) is isomorphic to Hn for some n. Spherical harmonics Hn :=the space of homogeneous harmonic polynomials of degree n in three variables. Harmonic means that they vanish under the Laplace ¶2 ¶2 ¶2 operator D = 2 + 2 + 2 . ¶x1 ¶x2 ¶x3 Theorem Hn is an irreducible representation of SO(3) 2 2 L¥ L (S ) = cn=0Hn, Spherical harmonics Hn :=the space of homogeneous harmonic polynomials of degree n in three variables. Harmonic means that they vanish under the Laplace ¶2 ¶2 ¶2 operator D = 2 + 2 + 2 . ¶x1 ¶x2 ¶x3 Theorem Hn is an irreducible representation of SO(3) 2 2 L¥ L (S ) = cn=0Hn, Every irreducible representation of SO(3) is isomorphic to Hn for some n. Spherical harmonics Hn :=the space of homogeneous harmonic polynomials of degree n in three variables. Harmonic means that they vanish under the Laplace ¶2 ¶2 ¶2 operator D = 2 + 2 + 2 .