Generalized Jucys-Murphy Elements and Canonical Idempotents in Brauer Algebras arXiv:1606.08900 Aaron Lauve Loyola University Chicago joint work with: Stephen Doty Loyola University Chicago George H. Seelinger University of Virginia SageDays@ICERM July 23–27, 2018
Plan of Talk / Motivation 1 Canonical Idempotents in multiplicity-free families of algebras 2 Wedderburn–Artin Theorem for tower of Brauer algebras 3 Module Decomposition for Doty’s Permutation modules Look for these boxes throughout. Sage Math Wish List For certain finite dimensional algebras: some_alg(smaller_alg) some_alg.centralizer(elt_lst) . . .
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Character theory dictates: equinumerous with the conj. classes in S r A simple calculation dictates: equinumerous with partitions ( λ $ r ) Where to look for λ ? Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 3 / 30
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Character theory dictates: equinumerous with the conj. classes in S r A simple calculation dictates: equinumerous with partitions ( λ $ r ) Where to look for λ ? Idea #1: Internally . . . S λ Ď S r Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 3 / 30
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Character theory dictates: equinumerous with the conj. classes in S r A simple calculation dictates: equinumerous with partitions ( λ $ r ) Where to look for λ ? Idea #1: Internally . . . S λ Ď S r Setup: k - field (char. p ě 0); V 0 - trivial rep. for S λ : “ S λ 1 ˆ S λ 2 ˆ ¨ ¨ ¨ ˆ S λ r Induce from the Young subgroup S λ Ď S r . Hey, look, a lambda! M λ : “ Ind S r “ V 0 b k S λ k S r . V 0 ˘ ` S λ Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 3 / 30
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Idea #2: Externally . . . weight space inside tensor space Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 4 / 30
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Idea #2: Externally . . . weight space inside tensor space Setup: k - field (char. p ě 0); V - vec. space over k (dim. n , w. basis t e j : 1 ď j ď n u ) Act on V b r by place permutation. E.g., ( n “ 4 , r “ 5), r e 3 b e 4 b e 3 b e 1 b e 2 s ˚ p 1 , 5 , 2 q “ r e 4 b e 2 b e 3 b e 1 b e 3 s . Focus on simple tensors of weight λ . E.g., wt p e 3 e 4 e 3 e 1 e 2 q “ p 1 , 1 , 2 , 1 q . Hey, look, a lambda? Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 4 / 30
Module Decomposition Permutation Modules for S r Let’s Study Irreducible Representations of S r Idea #2: Externally . . . weight space inside tensor space Setup: k - field (char. p ě 0); V - vec. space over k (dim. n , w. basis t e j : 1 ď j ď n u ) Act on V b r by place permutation. E.g., ( n “ 4 , r “ 5), r e 3 b e 4 b e 3 b e 1 b e 2 s ˚ p 1 , 5 , 2 q “ r e 4 b e 2 b e 3 b e 1 b e 3 s . Focus on simple tensors of weight λ . E.g., wt p e 3 e 4 e 3 e 1 e 2 q “ p 1 , 1 , 2 , 1 q . Hey, look, a lambda? M λ : “ span ˜ � e J : J P r n s r ; wt i p J q “ λ i ( . M λ “ E.g., for λ “ p 4 , 1 q , ˜ @ D e 11112 , e 11121 , e 11211 , e 12111 , e 21111 . Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 4 / 30
Module Decomposition Specht Modules for S r and B n p z q Let’s Study Irreducible Representations of S r Happy Coincidence: M λ » ˜ M λ . UnHappy Fact: the M λ are rarely irreducible (take char. k “ 0). Look inside for the irreducible (“Specht”) modules S λ . Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 5 / 30
Module Decomposition Specht Modules for S r and B n p z q Let’s Study Irreducible Representations of S r Happy Coincidence: M λ » ˜ M λ . UnHappy Fact: the M λ are rarely irreducible (take char. k “ 0). Look inside for the irreducible (“Specht”) modules S λ . Turning to Brauer algebras B n p z q . . . Hartmann–Paget (’06) use “Idea #1” to build permutation modules for B n p z q . ⊲ They find analogs of Specht and Young modules in this context. Doty (’12) uses “Idea #2” to build permutation modules for B n p z q . ⊲ We find Specht, and perhaps Young, modules in his context. Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 5 / 30
Interlude: Symmetric Group Algebras The wrong way to find idempotents The right way to find idempotents
Wedderburn–Artin Refresher for S n The Symmetric Group Algebra C S n A semisimple algebra – simples indexed by partitions λ $ n Wedderburn–Artin decomp. – C S n – À λ $ n M d λ p C q Example ( n “ 3) p 123 q p 23 q » fi ˚ ÿ ˚ ˚ α g g ? — ffi p 13 q e Ð Ñ — ffi ˚ ˚ g P S n — ffi – fl ˚ p 132 q p 12 q Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 7 / 30
Wedderburn–Artin Refresher for S n The Symmetric Group Algebra C S n A semisimple algebra – simples indexed by partitions λ $ n Wedderburn–Artin decomp. – C S n – À λ $ n M d λ p C q Example ( n “ 3) p 123 q p 23 q » fi 1 ÿ 1 α g g — ffi e p 13 q Ð Ñ — ffi 1 g P S n — ffi – fl 1 p 132 q p 12 q Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 7 / 30
Wedderburn–Artin Goals Notation & Goals Find (nice) formulas for: 1 ε p λ q – central idempotents ( identities for matrix blocks ). Unique. Example ( C S 3 ) » fi 1 “ ε p q ` ε p q ` ε p q 1 — ffi e Ø — ffi 1 — ffi – fl 1 Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 8 / 30
Wedderburn–Artin Goals Notation & Goals Find (nice) formulas for: 1 ε p λ q – central idempotents ( identities for matrix blocks ). Unique. 2 ε λ ii – primitive idempotents ( diagonal entries within blocks ). Not. Example ( C S 3 ) » fi 1 “ ε p q ` ε p q ` ε p q 1 — ffi e Ø — ffi 1 — ffi ` ˘ ` ˘ ` ˘ “ ` ε 11 ` ε 22 ` ε 11 ε 11 – fl 1 Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 8 / 30
Wedderburn–Artin Goals Notation & Goals Find (nice) formulas for: 1 ε p λ q – central idempotents ( identities for matrix blocks ). Unique. 2 ε λ ii – primitive idempotents ( diagonal entries within blocks ). Not. Example ( C S 3 ) » fi 1 “ ε p q ` ε p q ` ε p q 1 — ffi e Ø — ffi 1 1 — ffi ` ˘ ` ˘ ` ˘ “ ` ε 11 ` ε 22 ` ε 11 ε 11 – fl 1 3 ε λ ij – full set of d 2 λ block matrix units, ex. ε 21 not asking for these Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 8 / 30
Wedderburn–Artin Toward Goals 1 & 2 Theorem (Young, 1928) 1 The central idempotents for C S n are indexed by partitions of n . 2 The primitive idempotents for C S n are indexed by standard Young tableaux of size n . Example ( C S n ) ε p q “ e 1 2 3 ε p q “ e 1 2 ` e 1 3 ε p q “ e 1 3 2 2 3 Proof. T – defined via row- ( column-) (anti-)symmetrizers R T T ( C T T ). e T T T T Proof Idea – study intricate combinatorics of interactions between R T T and C S S . . . 15 pages(!) in Garsia’s notes [Gar] T S Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 9 / 30
Wedderburn–Artin Toward Goals 1 & 2 Theorem (Vershik–Okounkov, 1996) 1 Central idempotents for C S ‚ – indexed by nodes in Young’s lattice. 2 Primitive idempotents for C S ‚ – indexed by paths in Young’s lattice. ∅ Example ( C S n ) ε p q “ ε 1 2 ` ε 1 3 3 2 ε p q “ ε 1 2 3 ` ε 1 2 4 ` ε 1 3 4 4 3 2 . . . . . . . . . Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 10 / 30
Wedderburn–Artin Goals 1 & 2 Theorem (Vershik–Okounkov, 1996) 1 Central idempotents for C S ‚ – indexed by nodes in branching graph. 2 Primitive idempotents for C S ‚ – indexed by paths in branching graph. ∅ (Simple Restriction) Branching Graph S µ , Res S ` n ´ 1 S λ ˘ µ Ð λ ð ñ Hom ‰ 0 n S . . . . . . . . . Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 11 / 30
Wedderburn–Artin Goals 1 & 2 Theorem (Vershik–Okounkov, 1996; ... ) 1 Central idempotents for C S ‚ – indexed by nodes in branching graph. 2 Primitive idempotents for C S ‚ – = descending products of centrals. ∅ (Simple Restriction) Branching Graph S µ , Res S ` n ´ 1 S λ ˘ µ Ñ λ ð ñ Hom ‰ 0 n S Ex. ε 1 2 4 : “ ε p q ε p q ε p q ε p q 3 . . . . . . . . . Proof. Easy induction on n . Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 11 / 30
Wedderburn–Artin Goals 1 & 2 Sage Math Wish sage: S3 = SymmetricGroupAlgebra(QQ, 3) sage: S3.central_primitive_idempotent([2,1]) sage: S3.primitive_idempotent([[1,3], [2]]) Ditto for other (towers of) semisimple algebras. End Interlude. Aaron Lauve (Loyola Chicago) Idempotents for Brauer Algebras 26 Jul. 2018 12 / 30
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