LEADER 05041nam 22006373 450 001 9910957044203321 005 20231110230553.0 010 $a9781470465223 010 $a1470465221 035 $a(CKB)4100000011975370 035 $a(MiAaPQ)EBC6661108 035 $a(Au-PeEL)EBL6661108 035 $a(OCoLC)1256821396 035 $a(RPAM)22487637 035 $a(EXLCZ)994100000011975370 100 $a20210901d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aBounded Littlewood Identities 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (129 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.270 311 08$a9781470446901 311 08$a1470446901 320 $aIncludes bibliographical references. 327 $aCover -- Title page -- Acknowledgements -- Chapter 1. Introduction -- 1.1. Littlewood identities -- 1.2. Outline -- Chapter 2. Macdonald-Koornwinder theory -- 2.1. Partitions -- 2.2. Generalised -shifted factorials -- 2.3. Rogers-Szego? polynomials -- 2.4. Plethystic notation -- 2.5. Macdonald polynomials -- 2.6. Koornwinder polynomials -- 2.7. Macdonald-Koornwinder polynomials -- 2.8. Hall-Littlewood polynomials -- Chapter 3. Virtual Koornwinder integrals -- 3.1. Basic definitions -- 3.2. Closed-form evaluations-the Macdonald case -- 3.3. Closed-form evaluations-the Hall-Littlewood case -- Chapter 4. Bounded Littlewood identities -- 4.1. Statement of results -- 4.2. Proofs of Theorems 4.1-4.8 -- Chapter 5. Applications -- 5.1. Plane partitions -- 5.2. Character identities for affine Lie algebras -- 5.3. Rogers-Ramanujan identities -- 5.4. Quadratic transformations for Kaneko-Macdonald-type basic hypergeometric series -- Chapter 6. Open problems -- 6.1. Missing -analogues -- 6.2. Littlewood identities for near-rectangular partitions -- 6.3. Littlewood identities of Pfaffian type -- 6.4. Elliptic Littlewood identities -- 6.5. , -Littlewood-Richardson coefficients -- 6.6. Dyson-Macdonald-type identities -- Appendix A. The Weyl-Kac formula -- Appendix B. Limits of elliptic hypergeometric integrals -- Bibliography -- Back Cover. 330 $a"We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon's famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n,R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers-Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko-Macdonald-type basic hypergeometric series"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aOrthogonal polynomials 606 $aCombinatorial identities 606 $aCombinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations$2msc 606 $aCombinatorics -- Algebraic combinatorics -- Combinatorial aspects of representation theory$2msc 606 $aNonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras$2msc 606 $aSpecial functions -- Basic hypergeometric functions -- Basic hypergeometric functions associated with root systems$2msc 615 0$aOrthogonal polynomials. 615 0$aCombinatorial identities. 615 7$aCombinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations. 615 7$aCombinatorics -- Algebraic combinatorics -- Combinatorial aspects of representation theory. 615 7$aNonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. 615 7$aSpecial functions -- Basic hypergeometric functions -- Basic hypergeometric functions associated with root systems. 676 $a515/.55 686 $a05E05$a05E10$a17B67$a33D67$2msc 700 $aRains$b Eric M$0724889 701 $aWarnaar$b S. Ole$01802234 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910957044203321 996 $aBounded Littlewood Identities$94347807 997 $aUNINA