Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture

Abstract

We prove the classification of homomorphisms from the algebra of symmetric functions to $\mathbb {R}$ with non-negative values on Macdonald symmetric functions $P_{\lambda }$, which was conjectured by S. V. Kerov in 1992.

  • [ASW] Go to document M. Aissen, I. J. Schoenberg, and A. M. Whitney, "On the generating functions of totally positive sequences. I," J. Analyse Math., vol. 2, pp. 93-103, 1952.
    @ARTICLE{ASW,
      author = {Aissen, Michael and Schoenberg, I. J. and Whitney, A. M.},
      title = {On the generating functions of totally positive sequences. {I}},
      journal = {J. Analyse Math.},
      fjournal = {Journal d'Analyse Mathématique},
      volume = {2},
      year = {1952},
      pages = {93--103},
      issn = {0021-7670},
      mrclass = {27.0X},
      mrnumber = {0053174},
      mrreviewer = {E. Hille},
      doi = {10.1007/BF02786970},
      url = {https://doi.org/10.1007/BF02786970},
      zblnumber = {0049.17201},
      }
  • [B1] Go to document A. M. Borodin, "Limit Jordan normal form of large triangular matrices over a finite field," Funktsional. Anal. i Prilozhen., vol. 29, iss. 4, pp. 72-75, 1995.
    @ARTICLE{B1,
      author = {Borodin, A. M.},
      title = {Limit {J}ordan normal form of large triangular matrices over a finite field},
      journal = {Funktsional. Anal. i Prilozhen.},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Funktsional\cprime ny\u{i} Analiz i ego Prilozheniya},
      volume = {29},
      year = {1995},
      number = {4},
      pages = {72--75},
      issn = {0374-1990},
      mrclass = {11C20 (11K99)},
      mrnumber = {1375543},
      doi = {10.1007/BF01077476},
      url = {https://doi.org/10.1007/BF01077476},
      zblnumber = {0860.15009},
      }
  • [B2] Go to document A. M. Borodin, "The law of large numbers and the central limit theorem for the Jordan normal form of large triangular matrices over a finite field," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), vol. 240, iss. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, pp. 18-43, 290, 1997.
    @ARTICLE{B2,
      author = {Borodin, A. M.},
      title = {The law of large numbers and the central limit theorem for the {J}ordan normal form of large triangular matrices over a finite field},
      journal = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u{i} Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
      volume = {240},
      year = {1997},
      number = {Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2},
      pages = {18--43, 290},
      issn = {0373-2703},
      mrclass = {60F15 (11K99 15A33 60F05)},
      mrnumber = {1691635},
      mrreviewer = {Vadim A. Ka\u{i}manovich},
      doi = {10.1007/BF02175823},
      url = {https://doi.org/10.1007/BF02175823},
      zblnumber = {0952.60020},
      }
  • [BBW] A. Borodin, A. Bufetov, and M. Wheeler, Between the stochastic six vertex model and Hall-Littlewood processes, 2016.
    @MISC{BBW,
      author = {Borodin, Alexei and Bufetov, A. and Wheeler, M.},
      title = {Between the stochastic six vertex model and {H}all-{L}ittlewood processes},
      year = {2016},
      arxiv = {1611.09486},
      zblnumber = {},
      }
  • [BC] Go to document A. Borodin and I. Corwin, "Macdonald processes," Probab. Theory Related Fields, vol. 158, iss. 1-2, pp. 225-400, 2014.
    @ARTICLE{BC,
      author = {Borodin, Alexei and Corwin, Ivan},
      title = {Macdonald processes},
      journal = {Probab. Theory Related Fields},
      fjournal = {Probability Theory and Related Fields},
      volume = {158},
      year = {2014},
      number = {1-2},
      pages = {225--400},
      issn = {0178-8051},
      mrclass = {60K35 (33D52 60H15 82B23 82C22)},
      mrnumber = {3152785},
      mrreviewer = {Rongfeng Sun},
      doi = {10.1007/s00440-013-0482-3},
      url = {https://doi.org/10.1007/s00440-013-0482-3},
      zblnumber = {1291.82077},
      }
  • [BO] Go to document A. Borodin and G. Olshanski, Representations of the infinite symmetric group, Cambridge Univ. Press, Cambridge, 2017, vol. 160.
    @BOOK{BO,
      author = {Borodin, Alexei and Olshanski, Grigori},
      title = {Representations of the infinite symmetric group},
      series = {Cambridge Studies in Advanced Mathematics},
      volume = {160},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2017},
      pages = {vii+160},
      isbn = {978-1-107-17555-6},
      mrclass = {20C32 (05E10 05E45 22D10 60J50 60K35)},
      mrnumber = {3618143},
      mrreviewer = {Sevak Mkrtchyan},
      doi = {10.1017/CBO9781316798577},
      url = {https://doi.org/10.1017/CBO9781316798577},
      zblnumber = {1364.20001},
      }
  • [BuP] Go to document A. Bufetov and L. Petrov, "Law of large numbers for infinite random matrices over a finite field," Selecta Math. (N.S.), vol. 21, iss. 4, pp. 1271-1338, 2015.
    @ARTICLE{BuP,
      author = {Bufetov, Alexey and Petrov, Leonid},
      title = {Law of large numbers for infinite random matrices over a finite field},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {21},
      year = {2015},
      number = {4},
      pages = {1271--1338},
      issn = {1022-1824},
      mrclass = {60F15 (05E05 05E10 20G40 33C67 60B15 60K35 82C22)},
      mrnumber = {3397450},
      mrreviewer = {Sho Matsumoto},
      doi = {10.1007/s00029-015-0179-9},
      url = {https://doi.org/10.1007/s00029-015-0179-9},
      zblnumber = {1335.60032},
      }
  • [BuG] Go to document A. Bufetov and V. Gorin, "Stochastic monotonicity in Young graph and Thoma theorem," Int. Math. Res. Not. IMRN, iss. 23, pp. 12920-12940, 2015.
    @ARTICLE{BuG,
      author = {Bufetov, Alexey and Gorin, Vadim},
      title = {Stochastic monotonicity in {Y}oung graph and {T}homa theorem},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2015},
      number = {23},
      pages = {12920--12940},
      issn = {1073-7928},
      mrclass = {60C05 (05E05)},
      mrnumber = {3431640},
      mrreviewer = {Maciej Do\l \polhk ega},
      doi = {10.1093/imrn/rnv085},
      url = {https://doi.org/10.1093/imrn/rnv085},
      zblnumber = {1327.05337},
      }
  • [Edr] Go to document A. Edrei, "On the generating functions of totally positive sequences. II," J. Analyse Math., vol. 2, pp. 104-109, 1952.
    @ARTICLE{Edr,
      author = {Edrei, Albert},
      title = {On the generating functions of totally positive sequences. {II}},
      journal = {J. Analyse Math.},
      fjournal = {Journal d'Analyse Mathématique},
      volume = {2},
      year = {1952},
      pages = {104--109},
      issn = {0021-7670},
      mrclass = {27.0X},
      mrnumber = {0053175},
      mrreviewer = {E. Hille},
      doi = {10.1007/BF02786971},
      url = {https://doi.org/10.1007/BF02786971},
      zblnumber = {0049.17202},
      }
  • [Ful] Go to document W. Fulton, Young Tableaux, Cambridge Univ. Press, Cambridge, 1996, vol. 35.
    @BOOK{Ful,
      author = {Fulton, William},
      title = {Young Tableaux},
      series = {London Math. Soc. Student Texts},
      volume = {35},
      titlenote = {{W}ith {A}pplications to {R}epresentation {T}heory and {G}eometry},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {1996},
      pages = {x+260},
      isbn = {0-521-56144-2; 0-521-56724-6},
      mrclass = {05E10 (05E05 05E15 14M15 20G05)},
      mrnumber = {1464693},
      mrreviewer = {Tadeusz Józefiak},
      zblnumber = {0878.14034},
      doi = {10.1017/CBO9780511626241},
      }
  • [GK] Go to document F. P. Gantmacher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised ed., AMS Chelsea Publishing, Providence, RI, 2002.
    @BOOK{GK,
      author = {Gantmacher, F. P. and Krein, M. G.},
      title = {Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems},
      edition = {revised},
      note = {translation based on the 1941 Russian original, Edited and with a preface by Alex Eremenko},
      publisher = {AMS Chelsea Publishing, Providence, RI},
      year = {2002},
      pages = {viii+310},
      isbn = {0-8218-3171-2},
      mrclass = {34L05 (15A18 34A30 34B05 45B05 47G10 74H45)},
      mrnumber = {1908601},
      doi = {10.1090/chel/345},
      url = {https://doi.org/10.1090/chel/345},
      zblnumber = {1002.74002},
      }
  • [GO] Go to document A. Gnedin and G. Olshanski, "Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams," Int. Math. Res. Not., p. I, 2006.
    @ARTICLE{GO,
      author = {Gnedin, Alexander and Olshanski, Grigori},
      title = {Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      year = {2006},
      pages = {Art. ID 51968, 39},
      issn = {1073-7928},
      mrclass = {05E10 (05C05)},
      mrnumber = {2211157},
      mrreviewer = {Thomas Fun Yau Lam},
      doi = {10.1155/IMRN/2006/51968},
      url = {https://doi.org/10.1155/IMRN/2006/51968},
      zblnumber = {1102.05001},
      }
  • [GKV] Go to document V. Gorin, S. Kerov, and A. Vershik, "Finite traces and representations of the group of infinite matrices over a finite field," Adv. Math., vol. 254, pp. 331-395, 2014.
    @ARTICLE{GKV,
      author = {Gorin, Vadim and Kerov, Sergei and Vershik, Anatoly},
      title = {Finite traces and representations of the group of infinite matrices over a finite field},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {254},
      year = {2014},
      pages = {331--395},
      issn = {0001-8708},
      mrclass = {22D10 (20G40 22D15)},
      mrnumber = {3161102},
      mrreviewer = {C. Ryan Vinroot},
      doi = {10.1016/j.aim.2013.12.028},
      url = {https://doi.org/10.1016/j.aim.2013.12.028},
      zblnumber = {1286.22004},
      }
  • [Iv] Go to document V. N. Ivanov, "The dimension of skew-shifted Young diagrams, and projective characters of the infinite symmetric group," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), vol. 240, iss. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, pp. 115-135, 292, 1997.
    @ARTICLE{Iv,
      author = {Ivanov, V. N.},
      title = {The dimension of skew-shifted {Y}oung diagrams, and projective characters of the infinite symmetric group},
      journal = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u{i} Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
      volume = {240},
      year = {1997},
      number = {Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2},
      pages = {115--135, 292--293},
      issn = {0373-2703},
      mrclass = {05E05 (05E10 20C25 20C32)},
      mrnumber = {1691642},
      mrreviewer = {Tadeusz Józefiak},
      doi = {10.1007/BF02175830},
      url = {https://doi.org/10.1007/BF02175830},
      zblnumber = {0960.20009},
      }
  • [Kar] S. Karlin, Total Positivity. Vol. I, Stanford Univ. Press, Stanford, Calif, 1968.
    @BOOK{Kar,
      author = {Karlin, Samuel},
      title = {Total Positivity. {V}ol. {I}},
      publisher = {Stanford Univ. Press, Stanford, Calif},
      year = {1968},
      pages = {xii+576},
      mrclass = {46.00 (41.00)},
      mrnumber = {0230102},
      mrreviewer = {I. I. Hirschman, Jr.},
      zblnumber = {0219.47030},
      }
  • [Ker89] Go to document S. V. Kerov, "Combinatorial examples in the theory of AF-algebras," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 172, iss. Differentsial\cprime naya Geom. Gruppy Li i Mekh. Vol. 10, pp. 55-67, 169, 1989.
    @ARTICLE{Ker89,
      author = {Kerov, S. V.},
      title = {Combinatorial examples in the theory of {AF}-algebras},
      journal = {Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)},
      fjournal = {Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)},
      volume = {172},
      year = {1989},
      number = {Differentsial\cprime naya Geom. Gruppy Li i Mekh. Vol. 10},
      pages = {55--67, 169--170},
      issn = {0373-2703},
      mrclass = {46L80 (05A30 05C99 06F15 19K14)},
      mrnumber = {1015698},
      mrreviewer = {G. A. Elliott},
      doi = {10.1007/BF01480687},
      url = {https://doi.org/10.1007/BF01480687},
      zblnumber = {0747.46045},
      }
  • [Ker92] Go to document S. V. Kerov, "Generalized Hall-Littlewood symmetric functions and orthogonal polynomials," in Representation Theory and Dynamical Systems, Amer. Math. Soc., Providence, RI, 1992, vol. 9, pp. 67-94.
    @INCOLLECTION{Ker92,
      author = {Kerov, S. V.},
      title = {Generalized {H}all-{L}ittlewood symmetric functions and orthogonal polynomials},
      booktitle = {Representation Theory and Dynamical Systems},
      series = {Adv. Soviet Math.},
      volume = {9},
      pages = {67--94},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1992},
      mrclass = {05E05 (33C45)},
      mrnumber = {1166196},
      mrreviewer = {Tadeusz Józefiak},
      zblnumber = {0760.05089},
      doi = {10.1090/advsov/009/03},
      }
  • [Ker03] Go to document S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, American Mathematical Society, Providence, RI, 2003, vol. 219.
    @BOOK{Ker03,
      author = {Kerov, S. V.},
      title = {Asymptotic representation theory of the symmetric group and its applications in analysis},
      series = {Transl. Math. Monogr.},
      volume = {219},
      note = {translated from the Russian manuscript by N. V. Tsilevich, With a foreword by A. Vershik and comments by G. Olshanski},
      publisher = {American Mathematical Society, Providence, RI},
      year = {2003},
      pages = {xvi+201},
      isbn = {0-8218-3440-1},
      mrclass = {20C30 (05E05 05E10 20C32 20P05 60C05)},
      mrnumber = {1984868},
      mrreviewer = {Akihito Hora},
      zblnumber = {1031.20007},
      doi = {10.1090/mmono/219},
      }
  • [KOO] Go to document S. Kerov, A. Okounkov, and G. Olshanski, "The boundary of the Young graph with Jack edge multiplicities," Internat. Math. Res. Notices, iss. 4, pp. 173-199, 1998.
    @ARTICLE{KOO,
      author = {Kerov, Sergei and Okounkov, Andrei and Olshanski, Grigori},
      title = {The boundary of the {Y}oung graph with {J}ack edge multiplicities},
      journal = {Internat. Math. Res. Notices},
      fjournal = {International Mathematics Research Notices},
      year = {1998},
      number = {4},
      pages = {173--199},
      issn = {1073-7928},
      mrclass = {05E10 (20C30 31C35 33C80)},
      mrnumber = {1609628},
      mrreviewer = {Tim H. Baker},
      doi = {10.1155/S1073792898000154},
      url = {https://doi.org/10.1155/S1073792898000154},
      zblnumber = {0960.05107},
      }
  • [Kin] Go to document J. F. C. Kingman, "Random partitions in population genetics," Proc. Roy. Soc. London Ser. A, vol. 361, iss. 1704, pp. 1-20, 1978.
    @ARTICLE{Kin,
      author = {Kingman, J. F. C.},
      title = {Random partitions in population genetics},
      journal = {Proc. Roy. Soc. London Ser. A},
      fjournal = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences},
      volume = {361},
      year = {1978},
      number = {1704},
      pages = {1--20},
      issn = {0962-8444},
      mrclass = {92A15 (92A10)},
      mrnumber = {0526801},
      doi = {10.1098/rspa.1978.0089},
      url = {https://doi.org/10.1098/rspa.1978.0089},
      zblnumber = {0393.92011},
      }
  • [Mac] I. G. Macdonald, Symmetric Functions and Hall Polynomials, The Clarendon Press, Oxford Univ. Press, New York, 1979.
    @BOOK{Mac,
      author = {Macdonald, I. G.},
      title = {Symmetric Functions and {H}all Polynomials},
      note = {Oxford Math. Monogr.},
      publisher = {The Clarendon Press, Oxford Univ. Press, New York},
      year = {1979},
      pages = {viii+180},
      isbn = {0-19-853530-9},
      mrclass = {05-02 (12-02 20C30 20K01)},
      mrnumber = {0553598},
      mrreviewer = {Ira Gessel},
      zblnumber = {0487.20007},
      }
  • [Me] Go to document P. Méliot, Representation Theory of Symmetric Groups, CRC Press, Boca Raton, FL, 2017.
    @BOOK{Me,
      author = {Méliot, Pierre-Lo\"{i}c},
      title = {Representation Theory of Symmetric Groups},
      series = {Discrete Math. Appl. (Boca Raton)},
      publisher = {CRC Press, Boca Raton, FL},
      year = {2017},
      pages = {xvi+666},
      isbn = {978-1-4987-1912-4},
      mrclass = {20C30 (05E05 05E10 20B30 20C08 20C32 60B15 60F05)},
      mrnumber = {3616172},
      mrreviewer = {Mark J. Wildon},
      doi = {10.1201/9781315371016},
      url = {https://doi.org/10.1201/9781315371016},
      zblnumber = {06680366},
      }
  • [N] Go to document M. L. Nazarov, "Quotient representations of the infinite spin-symmetric group," Uspekhi Mat. Nauk, vol. 43, iss. 4(262), pp. 221-222, 1988.
    @ARTICLE{N,
      author = {Nazarov, M. L.},
      title = {Quotient representations of the infinite spin-symmetric group},
      journal = {Uspekhi Mat. Nauk},
      fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      volume = {43},
      year = {1988},
      number = {4(262)},
      pages = {221--222},
      issn = {0042-1316},
      mrclass = {20C32},
      mrnumber = {0969586},
      mrreviewer = {Horia Scutaru},
      doi = {10.1070/RM1988v043n04ABEH001912},
      url = {https://doi.org/10.1070/RM1988v043n04ABEH001912},
      zblnumber = {0699.20012},
      }
  • [Ok94] Go to document A. Okounkov, "Thoma’s theorem and representations of an infinite bisymmetric group," Funktsional. Anal. i Prilozhen., vol. 28, iss. 2, pp. 31-40, 95, 1994.
    @ARTICLE{Ok94,
      author = {Okounkov, Andrei},
      title = {Thoma's theorem and representations of an infinite bisymmetric group},
      journal = {Funktsional. Anal. i Prilozhen.},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Funktsional\cprime ny\u{i} Analiz i ego Prilozheniya},
      volume = {28},
      year = {1994},
      number = {2},
      pages = {31--40, 95},
      issn = {0374-1990},
      mrclass = {20C32},
      mrnumber = {1283250},
      doi = {10.1007/BF01076496},
      url = {https://doi.org/10.1007/BF01076496},
      zblnumber = {0830.20029},
      }
  • [Ok97] Go to document A. Okounkov, "On representations of the infinite symmetric group," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), vol. 240, iss. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2, pp. 166-228, 294, 1997.
    @ARTICLE{Ok97,
      author = {Okounkov, Andrei},
      title = {On representations of the infinite symmetric group},
      journal = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u{i} Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
      volume = {240},
      year = {1997},
      number = {Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2},
      pages = {166--228, 294},
      issn = {0373-2703},
      mrclass = {20C32 (20B30)},
      mrnumber = {1691646},
      doi = {10.1007/BF02175834},
      url = {https://doi.org/10.1007/BF02175834},
      zblnumber = {0894.05053},
      }
  • [Ok98] Go to document A. Okounkov, "(Shifted) Macdonald polynomials: $q$-integral representation and combinatorial formula," Compositio Math., vol. 112, iss. 2, pp. 147-182, 1998.
    @ARTICLE{Ok98,
      author = {Okounkov, Andrei},
      title = {({S}hifted) {M}acdonald polynomials: {$q$}-integral representation and combinatorial formula},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {112},
      year = {1998},
      number = {2},
      pages = {147--182},
      issn = {0010-437X},
      mrclass = {05E05},
      mrnumber = {1626029},
      mrreviewer = {Angèle M. Hamel},
      doi = {10.1023/A:1000436921311},
      url = {https://doi.org/10.1023/A:1000436921311},
      zblnumber = {0897.05085},
      }
  • [OO1] A. Okounkov and G. Olshanski, "Shifted Schur functions," Algebra i Analiz, vol. 9, iss. 2, pp. 73-146, 1997.
    @ARTICLE{OO1,
      author = {Okounkov, Andrei and Olshanski, G.},
      title = {Shifted {S}chur functions},
      journal = {Algebra i Analiz},
      fjournal = {Rossi\u{i}skaya Akademiya Nauk. Algebra i Analiz},
      volume = {9},
      year = {1997},
      number = {2},
      pages = {73--146},
      note={[English translation: {\em St. Petersburg Math. J.} {\bf 9} (1998), 239300. Available at \url{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=762&option_lang=eng}{}]},
      issn = {0234-0852},
      mrclass = {05E05 (05E10 17B35 20G05)},
      mrnumber = {1468548},
      mrreviewer = {Witold Kra\'{s}kiewicz},
      zblnumber = {0894.05053},
      }
  • [OO2] Go to document A. Okounkov and G. Olshanski, "Shifted Jack polynomials, binomial formula, and applications," Math. Res. Lett., vol. 4, iss. 1, pp. 69-78, 1997.
    @ARTICLE{OO2,
      author = {Okounkov, Andrei and Olshanski, G.},
      title = {Shifted {J}ack polynomials, binomial formula, and applications},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {4},
      year = {1997},
      number = {1},
      pages = {69--78},
      issn = {1073-2780},
      mrclass = {05E05 (33D80)},
      mrnumber = {1432811},
      mrreviewer = {Peter J. Forrester},
      doi = {10.4310/MRL.1997.v4.n1.a7},
      url = {https://doi.org/10.4310/MRL.1997.v4.n1.a7},
      zblnumber = {0995.33013},
      }
  • [Ol] Go to document G. Olshanski, "Unitary representations of $(G,K)$-pairs that are connected with the infinite symmetric group $S(\infty)$," Algebra i Analiz, vol. 1, iss. 4, pp. 178-209, 1989.
    @ARTICLE{Ol,
      author = {Olshanski, G.},
      title = {Unitary representations of {$(G,K)$}-pairs that are connected with the infinite symmetric group {$S(\infty)$}},
      journal = {Algebra i Analiz},
      fjournal = {Algebra i Analiz},
      volume = {1},
      year = {1989},
      number = {4},
      pages = {178--209},
      issn = {0234-0852},
      mrclass = {22A25 (20C32)},
      mrnumber = {1027466},
      zblnumber = {0731.20009},
      url = {http://mi.mathnet.ru/eng/aa/v1/i4/p178},
      }
  • [Pin] Go to document A. Pinkus, Totally Positive Matrices, Cambridge Univ. Press, Cambridge, 2010, vol. 181.
    @BOOK{Pin,
      author = {Pinkus, Allan},
      title = {Totally Positive Matrices},
      series = {Cambridge Tracts in Math.},
      volume = {181},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2010},
      pages = {xii+182},
      isbn = {978-0-521-19408-2},
      mrclass = {15B48},
      mrnumber = {2584277},
      mrreviewer = {Ronald L. Smith},
      zblnumber = {1185.15028},
      doi = {10.1017/CBO9780511691713},
      }
  • [Ra] Go to document A. Ram, "Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux," Pure Appl. Math. Q., vol. 2, iss. 4, Special Issue: In honor of Robert D. MacPherson. Part 2, pp. 963-1013, 2006.
    @ARTICLE{Ra,
      author = {Ram, Arun},
      title = {Alcove walks, {H}ecke algebras, spherical functions, crystals and column strict tableaux},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {2},
      year = {2006},
      number = {4, Special Issue: In honor of Robert D. MacPherson. Part 2},
      pages = {963--1013},
      issn = {1558-8599},
      mrclass = {20C08 (05E15 17B10 20G05 33D52)},
      mrnumber = {2282411},
      mrreviewer = {Guy Rousseau},
      doi = {10.4310/PAMQ.2006.v2.n4.a4},
      url = {https://doi.org/10.4310/PAMQ.2006.v2.n4.a4},
      zblnumber = {1127.20005},
      }
  • [Sc] Go to document C. Schwer, "Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra," Int. Math. Res. Not., p. I, 2006.
    @ARTICLE{Sc,
      author = {Schwer, Christoph},
      title = {Galleries, {H}all-{L}ittlewood polynomials, and structure constants of the spherical {H}ecke algebra},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      year = {2006},
      pages = {Art. ID 75395, 31},
      issn = {1073-7928},
      mrclass = {22E50 (05E05 17B20 20C08)},
      mrnumber = {2264725},
      mrreviewer = {Arun Ram},
      doi = {10.1155/IMRN/2006/75395},
      url = {https://doi.org/10.1155/IMRN/2006/75395},
      zblnumber = {1121.05121},
      }
  • [Sch48] I. J. Schoenberg, "Some analytical aspects of the problem of smoothing," in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 351-370.
    @INCOLLECTION{Sch48,
      author = {Schoenberg, I. J.},
      title = {Some analytical aspects of the problem of smoothing},
      booktitle = {Studies and {E}ssays {P}resented to {R}. {C}ourant on his 60th {B}irthday, {J}anuary 8, 1948},
      pages = {351--370},
      publisher = {Interscience Publishers, Inc., New York},
      year = {1948},
      mrclass = {27.0X},
      mrnumber = {0023309},
      mrreviewer = {T. N. E. Greville},
      zblnumber = {0033.06403},
      }
  • [Sch88] I. J. Schoenberg, Selected Papers. Vol. 1, Birkhäuser Boston, Inc., Boston, MA, 1988.
    @BOOK{Sch88,
      author = {Schoenberg, I. J.},
      title = {Selected Papers. {V}ol. 1},
      series = {Contemp. Math.},
      note = {with contributions by P. Erdős, J.-L. Goffin, S. Cambanis, D. Richards, R. Askey and S. Ruscheweyh; edited and with a foreword by Carl de Boor},
      publisher = {Birkhäuser Boston, Inc., Boston, MA},
      year = {1988},
      pages = {xviii+405},
      isbn = {0-8176-3404-5},
      mrclass = {01A75 (41A15)},
      mrnumber = {1015295},
      mrreviewer = {Weiyi Su},
      zblnumber = {0933.01032},
      }
  • [S] Go to document J. Schur, "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen," J. Reine Angew. Math., vol. 139, pp. 155-250, 1911.
    @ARTICLE{S,
      author = {Schur, J.},
      title = {Über die {D}arstellung der symmetrischen und der alternierenden {G}ruppe durch gebrochene lineare {S}ubstitutionen},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {139},
      year = {1911},
      pages = {155--250},
      issn = {0075-4102},
      mrclass = {DML},
      mrnumber = {1580818},
      doi = {10.1515/crll.1911.139.155},
      url = {https://doi.org/10.1515/crll.1911.139.155},
      zblnumber = {42.0154.02},
      }
  • [St] Go to document R. P. Stanley, "Some combinatorial properties of Jack symmetric functions," Adv. Math., vol. 77, iss. 1, pp. 76-115, 1989.
    @ARTICLE{St,
      author = {Stanley, Richard P.},
      title = {Some combinatorial properties of {J}ack symmetric functions},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {77},
      year = {1989},
      number = {1},
      pages = {76--115},
      issn = {0001-8708},
      mrclass = {05A15 (20C30)},
      mrnumber = {1014073},
      mrreviewer = {Dennis White},
      doi = {10.1016/0001-8708(89)90015-7},
      url = {https://doi.org/10.1016/0001-8708(89)90015-7},
      zblnumber = {0743.05072},
      }
  • [Th] Go to document E. Thoma, "Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe," Math. Z., vol. 85, pp. 40-61, 1964.
    @ARTICLE{Th,
      author = {Thoma, Elmar},
      title = {Die unzerlegbaren, positiv-definiten {K}lassenfunktionen der abzählbar unendlichen, symmetrischen {G}ruppe},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {85},
      year = {1964},
      pages = {40--61},
      issn = {0025-5874},
      mrclass = {46.65 (22.60)},
      mrnumber = {0173169},
      mrreviewer = {S. Sakai},
      doi = {10.1007/BF01114877},
      url = {https://doi.org/10.1007/BF01114877},
      zblnumber = {0192.12402},
      }
  • [V03] Go to document A. Vershik, "Two lectures on the asymptotic representation theory and statistics of Young diagrams," in Asymptotic Combinatorics with Applications to Mathematical Physics, Springer, Berlin, 2003, vol. 1815, pp. 161-182.
    @INCOLLECTION{V03,
      author = {Vershik, A.},
      title = {Two lectures on the asymptotic representation theory and statistics of {Y}oung diagrams},
      booktitle = {Asymptotic Combinatorics with Applications to Mathematical Physics},
      venue = {{S}t. {P}etersburg, 2001},
      series = {Lecture Notes in Math.},
      volume = {1815},
      pages = {161--182},
      publisher = {Springer, Berlin},
      year = {2003},
      mrclass = {05E10 (20C32 20D20 60C05)},
      mrnumber = {2009839},
      mrreviewer = {Akihito Hora},
      doi = {10.1007/3-540-44890-X_7},
      url = {https://doi.org/10.1007/3-540-44890-X_7},
      zblnumber = {1014.00010},
      }
  • [V14] Go to document A. M. Vershik, "The problem of describing central measures on the path spaces of graded graphs," Funct. Anal. Appl., vol. 48, iss. 4, pp. 256-271, 2014.
    @ARTICLE{V14,
      author = {Vershik, A. M.},
      title = {The problem of describing central measures on the path spaces of graded graphs},
      note = {Translation of Funktsional. Anal. i. Prilozhen. {{\bf{4}}8} (2014), no. 4, 26--46},
      journal = {Funct. Anal. Appl.},
      fjournal = {Functional Analysis and its Applications},
      volume = {48},
      year = {2014},
      number = {4},
      pages = {256--271},
      issn = {0016-2663},
      mrclass = {28D05 (46A55)},
      mrnumber = {3372738},
      mrreviewer = {S. I. Bezugly\u{i}},
      doi = {10.1007/s10688-014-0069-5},
      url = {https://doi.org/10.1007/s10688-014-0069-5},
      zblnumber = {1370.37005},
      }
  • [VK81] Go to document A. M. Vershik and S. V. Kerov, "Asymptotic theory of the characters of a symmetric group," Funktsional. Anal. i Prilozhen., vol. 15, iss. 4, pp. 15-27, 96, 1981.
    @ARTICLE{VK81,
      author = {Vershik, A. M. and Kerov, S. V.},
      title = {Asymptotic theory of the characters of a symmetric group},
      journal = {Funktsional. Anal. i Prilozhen.},
      fjournal = {Akademiya Nauk SSSR. Funktsional\cprime ny\u{i} Analiz i ego Prilozheniya},
      volume = {15},
      year = {1981},
      number = {4},
      pages = {15--27, 96},
      issn = {0374-1990},
      mrclass = {22D10 (22D20 46L99)},
      mrnumber = {0639197},
      mrreviewer = {G. L. Litvinov},
      zblnumber = {0507.20006},
      doi = {10.1007/BF01106153},
      }
  • [VK84] S. V. Kerov and A. M. Vershik, "Characters, factor representations and $K$-functor of the infinite symmetric group," in Operator Algebras and Group Representations, Vol. II, Pitman, Boston, MA, 1984, vol. 18, pp. 23-32.
    @INCOLLECTION{VK84,
      author = {Kerov, S. V. and Vershik, A. M.},
      title = {Characters, factor representations and {$K$}-functor of the infinite symmetric group},
      booktitle = {Operator Algebras and Group Representations, {V}ol. {II}},
      venue = {{N}eptun, 1980},
      series = {Monogr. Stud. Math.},
      volume = {18},
      pages = {23--32},
      publisher = {Pitman, Boston, MA},
      year = {1984},
      mrclass = {22D10 (16A54 20C32 22D40 46L80 46M20)},
      mrnumber = {0733300},
      mrreviewer = {E. Thoma},
      zblnumber = {0545.22001},
      }
  • [Whit] Go to document A. M. Whitney, "A reduction theorem for totally positive matrices," J. Analyse Math., vol. 2, pp. 88-92, 1952.
    @ARTICLE{Whit,
      author = {Whitney, A. M.},
      title = {A reduction theorem for totally positive matrices},
      journal = {J. Analyse Math.},
      fjournal = {Journal d'Analyse Mathématique},
      volume = {2},
      year = {1952},
      pages = {88--92},
      issn = {0021-7670},
      mrclass = {27.0X},
      mrnumber = {0053173},
      mrreviewer = {E. Hille},
      doi = {10.1007/BF02786969},
      url = {https://doi.org/10.1007/BF02786969},
      zblnumber = {0049.17104},
      }

Authors

Konstantin Matveev

Brandeis University, Waltham, MA