6 X 10.5 Three Line Title.P65

Cambridge University Press 978-0-521-11817-0 - Representation Theory of the Symmetric Groups: The Okounkov- Vershik Approach, Character Formulas, and Partition Algebras Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli Index More information

Index

∗-algebra, 315 Bell number, 391 Bernoulli numbers, 208 action box doubly transitive, 29 addable, 81 symmetric, 30 removable, 81 algebra, 14 branching graph ∗-anti-homomorphism, 15 of a multiplicity-free chain of groups, ∗-anti-isomorphism, 15 69 ∗-homomorphism, 15 of an inclusion of algebras, 342 ∗-isomorphism, 15 Bratteli diagram, 75, 342, 353 -sub, 341 Abelian, 14 centralizer of an algebra, 343 commutative, 14 character, 20 completely reducible, central, 267 338 virtual, 184 direct sum, 15 characteristic map, 189 ﬁnite dimensional, 15 class symmetric functions, 267 isomorphism, 334 column stabilizer of a tableau, 294 Jacobian radical, 335 commutative algebra, 14 left regular representation, 338 completely reducible algebra, 338 partition, 385, 392 composition, 126 right regular representation, 338 conjugate semisimple, 335 partition, 136 simple, 333, 335 convolution, 37 Temperley–Lieb, 390 Coxeter unital, 14 generator, 83 alphabet, 274 length, 83 alternating relations, 104 k-tensor, 366 cycle, 79 multilinear map, 367 cyclic vector, 10 operator, 365 representation, 3 dominance relation for partitions, antisymmetric 135 k-tensor, 366 multilinear map, 367 exponential generating function, axial distance, 112 208

409

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11817-0 - Representation Theory of the Symmetric Groups: The Okounkov- Vershik Approach, Character Formulas, and Partition Algebras Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli Index More information

410 Index

faithful representation, 318 idempotent Fourier transform, 25 primitive, 45 inversion formula, 45 inclusion matrix, 342 of an algebra, 336 induced operator, 348 of central functions, 25 intertwining operator, 4, 319 on L(G), 42 invariant subspace, 2, 318 Frobenius coordinates, 239 Jacobi, K. G. J., 193 function, 238 Jacobi–Trudi identity, 193, 195, 196 quotient, 238 dual, 193, 196, 197 function H-conjugacy invariant, 59 kernel of an algebra representation, K-invariant 318 bi-, 47 Kerov–Vershik formula, 271 left, right, 47 central, 20 lattice permutation, 274 class, 20 left regular representation of an algebra, conjugacy invariant, 20 338 lemma Garsia, A. M., 215, 218 Gelfand, 49 Gelfand pair, 31, 49 symmetric case, 30 symmetric, 31 Wielandt, 29 weakly symmetric, 49 length Gelfand–Tsetlin hook, 182 algebra, 71 of a partition, 157 basis, 70 tree, 309 matrix units, 331 reduced, 311 minimal central projection, 18 general linear group, 368 monoid, 385 linear representation, 369 partition, 385 character, 371 planar partition, 390 equivalence, 369 sub-, 387 invariant subspace, 369 multiplicity-free irreducible, 369 chain for a pair, 75 generalized chain of subgroups, 69 polytabloid, 295 for a pair, 75 Specht module, 296 representation, 17 Giambelli, G. Z., 193 subalgebra, 343 Greenhalgebras, 65 subgroup, 64 group multiset, 361 algebra, 37 of GL(V ), 374 orbit ambivalent, 50 symmetric, 30 general linear, 368 symmetric, 3 partition, 80 algebra, 385 Hasse diagram, 90 conjugate, 136 hook, 116 dominance relation, 135 height, 116 lenght, 157 length, 182 monoid, 385 skew, 130 propagating number, 387 height, 130 special planar, 389

Index 411

partition algebra, 392 multiplicity, 321 permutation restriction, 341 conjugate, 80 tensor product, 324 cycle, 79 trivial, 318 cycle structure, 80 right regular representation marked, 93 of an algebra, 338 planar rim hook tableau, 133 partition monoid, 390 row stabilizer of a tableau, 294 Plancherel formula, 45 polytabloid Schur’s lemma, 11 generalized, 295 seesaw pair, 345 primitive idempotent, 45 semistandard tableau, 145 propagating number of a partition, 387 shape of a Young frame, 81 shifted symmetric function, 237 Radon transform, 28, 145 simple algebra, 333 representation simple tensor, 8, 358 of GL(V ), 369 skew character, 371 hook, 130 equivalence, 369 height, 130 invariant subspace, 369 representation, 124 irreducible, 369 shape, 121 of a group, 1 connected, 122 K-invariant vectors, 19 disconnected, 122 conjugate, 6 totally disconnected, 122 adjoint, 6 skew-symmetric alternating, 3 multilinear map, 367 character, 20 Specht module, 296 commutant, 16 generalized, 296 contragredient, 6 special planar partition, 389 direct sum, 5 spectrum of YJM elements, 102 equivalence, 4 stabilizer, 30 extension, 66 Stirling numbers, 211 irreducible, 2 subalgebra, 341 isotypic component, 13, 321 submonoid, 387 left regular, 3 symmetric multiplicity, 13 k-tensor, 362 multiplicity-free, 17 group, 3 permutation, 2 multilinear map, 362 restriction, 2 orbit, 30 right regular, 3 power, 362 skew, 124 symmetrizing operator, 361 sub-, 2 trivial, 2 tableau unitarizable, 3 associated with a composition, unitary, 2 293 unitary equivalence, 4 column stabilizer, 294 of an algebra, 318 rim hook, 133 ∗-, 320 row stabilizer, 294 commutant, 320 semistandard, 145 faithful, 318 tabloid, 294 irreducible, 318 tag, 93 kernel, 318 Temperley–Lieb algebra, 390

412 Index

tensor product weight of a Young basis vector, 101 of operators, 9 word, 274 of representations internal, 10 YJM elements, 92 outer, 9 Young of vector spaces, 8 (branching) graph, 90 tower of algebras, 353, 393 basis, 101, 126 trace, 19 diagram transform hook, 116 Radon, 28 frame, 81 transposition covers, 90 adjacent admissible, 89 poset adjacent, 83 n-truncated, 381 admissible, 83, 104, 123 reduced, 148 Trudi, N., 193 seminormal units, 119 Jacobi–Trudi identity, 193, 195, 196 subgroup, 126 dual, 193, 196, 197 tableau, 81 content of, 85, 123 unit, 14 standard, 82 Young–Jucys–Murphy elements, 92 virtual character, 184 spectrum, 102