representation theory and geometry Geordie Williamson University of Sydney http://www.maths.usyd.edu.au/u/geordie/ICM.pdf representations We obtain a representation of our group of symmetries ρ : G Ñ GLpVq: representations ¨ ˛ ˚ ‹ Sym ˝ ‚Ă GLpR3q: 2 representations ¨ ˛ ˚ ‹ Sym ˝ ‚Ă GLpR3q: We obtain a representation of our group of symmetries ρ : G Ñ GLpVq: 2 representations e2 ´e1 ´ e2 e1 #˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸+ 1 0 1 ´1 ´1 0 0 ´1 ´1 1 0 1 ; ; ; ; ; 0 1 0 ´1 ´1 1 1 ´1 ´1 0 1 0 3 Symmetric group Sn The problem of understanding „ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation is hard. The theory of representations of Sn is rich, highly-developed and useful. Galois representations The passage(s) tvarieties{Qu ÝÑ tGalois representationsu is one of the most powerful tools of modern number theory. why study representations? 4 The theory of representations of Sn is rich, highly-developed and useful. Galois representations The passage(s) tvarieties{Qu ÝÑ tGalois representationsu is one of the most powerful tools of modern number theory. why study representations? Symmetric group Sn The problem of understanding „ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation is hard. 4 Galois representations The passage(s) tvarieties{Qu ÝÑ tGalois representationsu is one of the most powerful tools of modern number theory. why study representations? Symmetric group Sn The problem of understanding „ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation is hard. The theory of representations of Sn is rich, highly-developed and useful. 4 why study representations? Symmetric group Sn The problem of understanding „ tSn-setsu{isomorphism Ø tsubgroups of Snu{conjugation is hard. The theory of representations of Sn is rich, highly-developed and useful. Galois representations The passage(s) tvarieties{Qu ÝÑ tGalois representationsu is one of the most powerful tools of modern number theory. 4 There are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: R3 “ L ` H “ trivial ` A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations. simple representations Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. 5 R3 “ L ` H “ trivial ` A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations. simple representations Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: 5 A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations. simple representations Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: R3 “ L ` H “ trivial ` 5 A representation is semi-simple if it is isomorphic to a direct sum of simple representations. simple representations Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: R3 “ L ` H “ trivial ` A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. 5 simple representations Example Consider the symmetric group S3. It acts via permutation of coordinates on R3. There are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: R3 “ L ` H “ trivial ` A representation V of a group G is simple or irreducible if its only G-invariant subspaces are t0u and V. A representation is semi-simple if it is isomorphic to a direct sum of simple representations. 5 As before, there are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: However now L Ă H because 3 “ 0. We obtain a composition series F3 0 Ă L Ă H Ă 3: trivial We write (“Grothendieck group”, “multiplicities”) F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns: trivial jordan-hölder theorem Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the finite field with 3 elements.) 6 However now L Ă H because 3 “ 0. We obtain a composition series F3 0 Ă L Ă H Ă 3: trivial We write (“Grothendieck group”, “multiplicities”) F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns: trivial jordan-hölder theorem Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the finite field with 3 elements.) As before, there are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: 6 We obtain a composition series F3 0 Ă L Ă H Ă 3: trivial We write (“Grothendieck group”, “multiplicities”) F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns: trivial jordan-hölder theorem Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the finite field with 3 elements.) As before, there are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: However now L Ă H because 3 “ 0. 6 trivial We write (“Grothendieck group”, “multiplicities”) F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns: trivial jordan-hölder theorem Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the finite field with 3 elements.) As before, there are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: However now L Ă H because 3 “ 0. We obtain a composition series F3 0 Ă L Ă H Ă 3: 6 jordan-hölder theorem Example Consider the symmetric group S3. It acts via permutation of F3 F Z Z coordinates on 3. (Here 3 “ {3 is the finite field with 3 elements.) As before, there are two invariant subspaces: L “ tall coordinates equalu; H “ tcoordinates sum to zerou: However now L Ă H because 3 “ 0. We obtain a composition series F3 0 Ă L Ă H Ă 3: trivial We write (“Grothendieck group”, “multiplicities”) F3 F3 sign r 3s “ rLs ` rH{Ls ` r 3{Hs “ 2rtrivials ` rsigns: trivial 6 We search for a classification (“periodic table”), character formulas (“mass”, “number of neutrons”), … simple representations representations Ø “matter” ( simple representations Ø “elements” ; ; semi-simple Ø “elements don’t interract” 7 simple representations representations Ø “matter” ( simple representations Ø “elements” ; ; semi-simple Ø “elements don’t interract” We search for a classification (“periodic table”), character formulas (“mass”, “number of neutrons”), … 7 ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups… plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). 8 “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups… plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). 8 ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups… plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). 8 ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups… plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). 8 Related situations: non-compact Lie groups, p-adic groups… plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). 8 plan of talk “Semi-simple world” ‚ Finite groups: Maschke’s theorem (1897). ‚ Compact Lie groups: Weyl’s theorem (1925). “Beyond the semi-simple world” ‚ Infinite-dimensional representations of Lie algebras: Jantzen conjecture and Kazhdan-Lusztig conjecture (1979). ‚ Modular representations of reductive algebraic groups: Lusztig conjecture (1980) and new character formula (2018). ‚ Modular representations of symmetric groups: Billiards conjecture (2017). Related situations: non-compact Lie groups, p-adic groups… 8 geometric structure invariant forms symmetric, hermitian, … representation theory geometry Related feature: (hidden) semi-simplicity A geometric structure on a real (resp.