Alphadeterminant cyclic modules and Jacobi polynomials
Authors:
Kazufumi Kimoto, Sho Matsumoto and Masato Wakayama; with an appendix by Kazufumi Kimoto
Journal:
Trans. Amer. Math. Soc. 361 (2009), 64476473
MSC (2000):
Primary 22E47, 33C45; Secondary 43A90, 13A50
Published electronically:
July 14, 2009
MathSciNet review:
2538600
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For positive integers and , we study the cyclic module generated by the th power of the determinant . This cyclic module is isomorphic to the th tensor space of the symmetric th tensor space of for all but finitely many exceptional values of . If is exceptional, then the cyclic module is equivalent to a proper submodule of , i.e. the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in . The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and whose entries are polynomials in with rational coefficients. In particular, we determine the matrix completely when . In this case, the matrix becomes a scalar and is essentially given by a classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary. In the Appendix, we consider a variation of the spherical Fourier transformation for as a main tool for analyzing the same problems, and describe the case where by using the zonal spherical functions of the Gelfand pair .
 1.
George
E. Andrews, Richard
Askey, and Ranjan
Roy, Special functions, Encyclopedia of Mathematics and its
Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
(2000g:33001)
 2.
Susumu
Ariki, Junichi
Matsuzawa, and Itaru
Terada, Representations of Weyl groups on zero weight spaces of
𝔤modules, Algebraic and topological theories (Kinosaki, 1984)
Kinokuniya, Tokyo, 1986, pp. 546–568. MR
1102274
 3.
Eiichi
Bannai and Tatsuro
Ito, Algebraic combinatorics. I, The Benjamin/Cummings
Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
(87m:05001)
 4.
William
Fulton and Joe
Harris, Representation theory, Graduate Texts in Mathematics,
vol. 129, SpringerVerlag, New York, 1991. A first course; Readings in
Mathematics. MR
1153249 (93a:20069)
 5.
I.
Gelfand and M.
Neumark, Unitary representations of the Lorentz group, Acad.
Sci. USSR. J. Phys. 10 (1946), 93–94. MR 0017282
(8,132b)
 6.
Ronald
L. Graham, Donald
E. Knuth, and Oren
Patashnik, Concrete mathematics, 2nd ed., AddisonWesley
Publishing Company, Reading, MA, 1994. A foundation for computer science.
MR
1397498 (97d:68003)
 7.
K. Kimoto, Generalized content polynomials toward determinant cyclic modules, Preprint (2007).
 8.
Kazufumi
Kimoto and Masato
Wakayama, Invariant theory for singular
𝛼determinants, J. Combin. Theory Ser. A 115
(2008), no. 1, 1–31. MR 2378855
(2008m:15019), http://dx.doi.org/10.1016/j.jcta.2007.03.008
 9.
Kazufumi
Kimoto and Masato
Wakayama, Quantum 𝛼determinant cyclic modules of
𝒰_{𝓆}(𝔤𝔩_{𝔫}), J. Algebra
313 (2007), no. 2, 922–956. MR 2329577
(2008d:17021), http://dx.doi.org/10.1016/j.jalgebra.2006.12.015
 10.
I.
G. Macdonald, Symmetric functions and Hall polynomials, 2nd
ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University
Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science
Publications. MR
1354144 (96h:05207)
 11.
Sho
Matsumoto, 𝛼Pfaffian, Pfaffian point process and shifted
Schur measure, Linear Algebra Appl. 403 (2005),
369–398. MR 2140292
(2006d:15014), http://dx.doi.org/10.1016/j.laa.2005.02.005
 12.
Sho
Matsumoto and Masato
Wakayama, Alphadeterminant cyclic modules of
𝔤𝔩_{𝔫}(ℂ), J. Lie Theory
16 (2006), no. 2, 393–405. MR 2197599
(2007a:17011)
 13.
Tomoyuki
Shirai and Yoichiro
Takahashi, Random point fields associated with certain Fredholm
determinants. I. Fermion, Poisson and boson point processes, J. Funct.
Anal. 205 (2003), no. 2, 414–463. MR 2018415
(2004m:60104), http://dx.doi.org/10.1016/S00221236(03)00171X
 14.
Sergei
Yu. Slavyanov and Wolfgang
Lay, Special functions, Oxford Mathematical Monographs, Oxford
University Press, Oxford, 2000. A unified theory based on singularities;
With a foreword by Alfred Seeger; Oxford Science Publications. MR 1858237
(2004e:33003)
 15.
Gábor
Szegő, Orthogonal polynomials, 4th ed., American
Mathematical Society, Providence, R.I., 1975. American Mathematical
Society, Colloquium Publications, Vol. XXIII. MR 0372517
(51 #8724)
 16.
D.
VereJones, A generalization of permanents and determinants,
Linear Algebra Appl. 111 (1988), 119–124. MR 974048
(89j:15014), http://dx.doi.org/10.1016/00243795(88)900535
 17.
Hermann
Weyl, The classical groups, Princeton Landmarks in
Mathematics, Princeton University Press, Princeton, NJ, 1997. Their
invariants and representations; Fifteenth printing; Princeton Paperbacks.
MR
1488158 (98k:01049)
 1.
 G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
 2.
 S. Ariki, J. Matsuzawa and I. Terada, Representation of Weyl groups on zero weight spaces of modules, Algebraic and Topological Theories (Kinosaki, 1984), 546568, Kinokuniya, Tokyo, 1986. MR 1102274
 3.
 E. Bannai and T. Ito, Algebraic Combinatorics I, Association Schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. MR 0882540 (87m:05001)
 4.
 W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics 129, Readings in Mathematics, SpringerVerlag, New York, 1991. MR 1153249 (93a:20069)
 5.
 I. M. Gel'fand and M. A. Naĭmark, Unitary representations of the Lorentz group, Acad. Sci. USSR J. Phys. 10 (1946), 9394; Izvestiya Akad. Nauk SSSR Ser. Mat. 11 (1947), 411504. MR 0017282 (8:132b)
 6.
 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A foundation for computer science, Second edition, AddisonWesley Publishing Company, Reading, MA, 1994. MR 1397498 (97d:68003)
 7.
 K. Kimoto, Generalized content polynomials toward determinant cyclic modules, Preprint (2007).
 8.
 K. Kimoto and M. Wakayama, Invariant theory for singular determinants, J. Combin. Theory Ser. A 115 (2008), no. 1, 131. MR 2378855
 9.
 , Quantum determinant cyclic modules of , J. Algebra 313 (2007), 922956. MR 2329577 (2008d:17021)
 10.
 I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn., Oxford University Press, 1995. MR 1354144 (96h:05207)
 11.
 S. Matsumoto, Alphapfaffian, pfaffian point process and shifted Schur measure, Linear Algebra Appl. 403 (2005), 369398. MR 2140292 (2006d:15014)
 12.
 S. Matsumoto and M. Wakayama, Alphadeterminant cyclic modules of , J. Lie Theory 16 (2006), 393405. MR 2197599 (2007a:17011)
 13.
 T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes, J. Funct. Anal. 205 (2003), 414463. MR 2018415 (2004m:60104)
 14.
 S. Yu. Slavyanov and W. Lay, Special Functions  A Unified Theory Based on Singularities, Oxford Univ. Press, Oxford, 2000. MR 1858237 (2004e:33003)
 15.
 G. Szegö, Orthogonal Polynomials, 4th edn., American Mathematical Society, 1975. MR 0372517 (51:8724)
 16.
 D. VereJones, A generalization of permanents and determinants, Linear Algebra Appl. 111 (1988), 119124. MR 0974048 (89j:15014)
 17.
 H. Weyl, The Classical Groups: Their Invariants and Representations, Fifteenth printing, Princeton Landmarks in Mathematics, Princeton University Press, 1997. MR 1488158 (98k:01049)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
22E47,
33C45,
43A90,
13A50
Retrieve articles in all journals
with MSC (2000):
22E47,
33C45,
43A90,
13A50
Additional Information
Kazufumi Kimoto
Affiliation:
Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 9030213, Japan
Email:
kimoto@math.uryukyu.ac.jp
Sho Matsumoto
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 8128581, Japan
Address at time of publication:
Department of Mathematics, Nagoya University, Chikusa, Nagoya 4648602, Japan
Email:
shomatsumoto@math.nagoyau.ac.jp
Masato Wakayama
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 8128581, Japan
Email:
wakayama@math.kyushuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994709048600
PII:
S 00029947(09)048600
Keywords:
Alphadeterminant,
cyclic modules,
Jacobi polynomials,
singly confluent Heun ODE,
permanent,
Kostka numbers,
irreducible decomposition,
spherical Fourier transformation,
zonal spherical functions,
Gelfand pair
Received by editor(s):
October 29, 2007
Published electronically:
July 14, 2009
Additional Notes:
The second author was partially supported by GrantinAid for JSPS Fellows No. 17006193.
The third author was partially supported by GrantinAid for Exploratory Research No. 18654005.
Dedicated:
Dedicated to Professor Masaaki Yoshida on his sixtieth birthday.
Article copyright:
© Copyright 2009
American Mathematical Society
