[PDF]
http://dx.doi.org/10.3952/lithjphys.51101
Open access article / Atviros prieigos straipsnis
Lith. J. Phys. 51, 5–18 (2011)
AN EXPLICIT BASIS OF LOWERING
OPERATORS FOR IRREDUCIBLE REPRESENTATIONS OF UNITARY GROUPS
D.S. Sage and L. Smolinsky
Department of Mathematics, Louisiana State University, Baton
Rouge, LA 70803, USA
E-mail: sage@math.lsu.edu, smolinsk@math.lsu.edu
Received 2 July 2010; revised 13
December 2010; accepted 15 December 2010
The representation theory of the
unitary groups is of fundamental significance in many areas of
physics and chemistry. In order to label states in a physical
system with unitary symmetry, it is necessary to have explicit
bases for the irreducible representations. One systematic way of
obtaining bases is to generalize the ladder operator approach to
the representations of SU(2) by using the formalism of lowering
operators. Here, one identifies a basis for the algebra of all
lowering operators and, for each irreducible representation, gives
a prescription for choosing a subcollection of lowering operators
that yields a basis upon application to the highest weight vector.
Bases obtained through lowering operators are particularly
convenient for computing matrix coefficients of observables as the
calculations reduce to the commutation relations for the standard
matrix units. The best known examples of this approach are the
extremal projector construction of the Gelfand–Zetlin basis and
the crystal (or canonical) bases of Kashiwara and Lusztig. In this
paper, we describe another simple method of obtaining bases for
the irreducible representations via lowering operators. These
bases do not have the algebraic canonicity of the Gelfand–Zetlin
and crystal bases, but the combinatorics involved are much more
straightforward, making the bases particularly suited for physical
applications.
Keywords: unitary group, special unitary
group, irreducible representations, lowering operators, spin-free
quantum chemistry, many-body problem
PACS: 02.20.-a, 31.15.xh
ŽEMINANČIŲJŲ OPERATORIŲ, SKIRTŲ
UNITARINIŲ GRUPIŲ NEREDUKUOTINIAMS ATVAIZDAMS, IŠREIKŠTINĖ BAZĖ
D.S. Sage, L. Smolinsky
Luizianos valstijos universitetas, Baton Ružas, JAV
Unitarinių grupių atvaizdų teorija
fundamentaliai svarbi daugelyje fizikos ir chemijos sričių.
Unitarinės simetrijos fizikinės sistemos būsenų žymėjimui reikia
turėti išreikštines bazes neredukuotiniams atvaizdams. Vienas
sisteminių būdų gauti bazes yra apibendrinti laiptinių operatorių
metodą SU(2) atvaizdams, panaudojant žeminančiųjų operatorių
formalizmą. Čia nustatoma bazė visų žeminančiųjų operatorių
algebrai ir kiekvienam neredukuotiniam atvaizdui pateikiama
instrukcija, kaip parinkti žeminančiųjų operatorių rinkinio dalį,
kurią naudojant bazė gaunama iš didžiausio svorio vektoriaus.
Bazės, gautos žeminančiaisiais operatoriais, ypač patogios
skaičiuojant stebimų dydžių matricinius koeficientus, kadangi jie
virsta komutacijos sąryšiais standartiniams matriciniams
vienetams. Žinomiausi šito metodo pavyzdžiai yra Gelfando ir
Cetlino bazės sukonstravimas naudojant kraštutinius projektorius
bei kristalinės (arba kanoninės) Kašivaros ir Lustigo bazės.
Straipsnyje aprašomas kitas paprastas būdas gauti neredukuotinių
atvaizdų bazes naudojant žeminančiuosius operatorius. Šios bazės
nepasižymi Gelfando ir Cetlino ar kristalinių bazių kanoniškumu,
tačiau kombinatorika, su kuria susiduriama, yra daug paprastesnė
ir dėl to šios bazės ypač tinka fizikiniams taikymams.
References / Nuorodos
[1] L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum
Physics (Addison-Wesley, Reading, 1981),
http://www.amazon.com/Angular-Momentum-Quantum-Physics-Application/dp/0201135078
[2] A. Arima, Elliott’s SU(3) model and its developments in nuclear
physics, J. Phys. G 25, 581–588 (1999),
http://dx.doi.org/10.1088/0954-3899/25/4/003
[3] J.P. Elliott, Collective motion in the nuclear shell model I.
Classification schemes for states of mixed configurations, Proc. R.
Soc. London Ser. A 245, 128–145 (1958),
http://dx.doi.org/10.1098/rspa.1958.0072
[4] J.P. Elliott, Collective motion in the nuclear shell model II.
The introduction of intrinsic wave-functions, Proc. R. Soc. London
Ser. A 245, 562–581 (1958),
http://dx.doi.org/10.1098/rspa.1958.0101
[5] H. Georgi, Lie Algebras in Particle Physics, 2nd ed.
(Perseus Books, Reading, 1999),
http://www.amazon.co.uk/Lie-Algebras-Particle-Physics-Frontiers/dp/0738202339/
[6] M. Moshinsky, Group Theory and the Many-Body Problem
(Gordon and Breach, New York, 1968),
http://www.amazon.co.uk/Theory-Problem-Documents-Modern-Physics/dp/0677017405/
[7] J. Paldus, Unitary group approach to many-electron correlation
problem, in: The Unitary Group, Lecture Notes in Chemistry,
Vol. 22 (Springer-Verlag, New York, 1981) pp. 1–50,
http://www.amazon.co.uk/Unitary-Evaluation-Electronic-Energy-Elements/dp/3540102876/
[8] The Unitary Group, ed. J. Hinze, Lecture Notes in
Chemistry, Vol. 22 (Springer-Verlag, New York, 1981),
http://www.amazon.co.uk/Unitary-Evaluation-Electronic-Energy-Elements/dp/3540102876/
[9] F.A. Matsen and R. Pauncz, The Unitary Group in Quantum
Chemistry (Elsevier, Amsterdam, 1986),
http://www.amazon.com/Unitary-Quantum-Chemistry-Physical-Theoretical/dp/0444427309/
[10] I. Bengtsson and K. Zyczkowski, Geometry of Quantum States:
An Introduction to Quantum Entanglement (Cambridge University
Press, Cambridge, 2008),
http://www.amazon.com/Geometry-Quantum-States-Introduction-Entanglement/dp/052189140X/
[11] D. Uskov and A.R.P. Rau, Geometric phases and Bloch-sphere
constructions for SU(N) groups with a complete description of
the SU(4) group, Phys. Rev. A 78, 022331 (2008),
http://dx.doi.org/10.1103/PhysRevA.78.022331
[12] M. Moshinsky, Bases for the irreducible representations of the
unitary groups and some applications, J. Math. Phys. 4,
1128–1139 (1963),
http://dx.doi.org/10.1063/1.1704043
[13] F.A. Matsen, Scientific reminiscences, Int. J. Quant. Chem. 41,
7–14 (1992),
http://dx.doi.org/10.1002/qua.560410105
[14] I. Schur, Über die rationalen Darstellungen der allgemeinen
linearen Gruppe, 1927, in: I. Schur, Gesammelte Abhandlungen III
(Springer, Berlin, 1973) pp. 68–85,
http://www.amazon.com/Gesammelte-Abhandlungen-Issai-Schur/dp/0387056300/
[15] H. Weyl, The Classical Groups (Princeton University
Press, Princeton, NJ, 1997),
http://www.amazon.com/Classical-Groups-Their-Invariants-Representations/dp/0691057567/
[16] H.Weyl, The Theory of Groups and Quantum Mechanics
(Dutton, New York, 1931),
http://www.amazon.com/Theory-Groups-Quantum-Mechanics-Hermann/dp/B001THPJ2C/
[17] J.G. Nagel and M. Moshinsky, Operators that lower or raise the
irreducible vector spaces of Un−1 contained in an
irreducible vector space of Un, J. Math. Phys. 6,
682–694 (1965),
http://dx.doi.org/10.1063/1.1704326
[18] R.M. Asherova, Yu.F. Smirnov, and V.N. Tolstoi, Projection
operators for simple Lie groups, Teoret. Mat. Fiz. 8,
255–271 (1971) [in Russian; English translation: Theoret. Math.
Phys. 8, 813–825 (1971)],
http://dx.doi.org/10.1007/BF01038003
[19] R.M. Asherova, Yu.F. Smirnov, and V.N. Tolstoi, Projection
operators for simple Lie groups II. General scheme for constructing
lowering operators. The groups SU(n), Teoret. Mat.
Fiz. 15, 107–119 (1973) [in Russian; English translation:
Theoret. Math. Phys. 15, 392–401 (1973)],
http://dx.doi.org/10.1007/BF01028268
[20] D.P. Zhelobenko, Extremal projectors and generalized Mickelsson
algebras on reductive Lie algebras, Izv. Akad. Nauk SSSR Ser. Mat. 52,
758–773, 895 (1988) [in Russian; English translation: Math. USSR
Izv. 33, 85–100 (1989),
http://dx.doi.org/10.1070/IM1989v033n01ABEH000815
[21] S. Khoroshkin and O. Ogievetsky, Mickelsson algebras and
Zhelobenko operators, J. Algebra 319, 2113–2165 (2008),
http://dx.doi.org/10.1016/j.jalgebra.2007.04.020
[22] J. Mickelsson, Step algebras of semi-simple subalgebras of Lie
algebras, Rep. Math. Phys. 4, 307–318 (1973),
http://dx.doi.org/10.1016/0034-4877(73)90006-2
[23] D.P. Zhelobenko, S-algebras and Verma modules over reductive
Lie algebras, Soviet Math. Dokl. 28, 696–700 (1983)
[24] A.I. Molev, Gelfand–Tsetlin bases for classical Lie algebras,
in: Handbook of Algebra, Vol. 4, ed. M. Hazewinkel
(Elsevier, 2006) pp. 109–170,
http://www.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-52213-9#description
[25] A.I. Molev, Weight bases of Gelfand–Tsetlin type for
representations of classical Lie algebras, J. Phys. A 33,
4143–4168 (2000),
http://dx.doi.org/10.1088/0305-4470/33/22/316
[26] V.N. Tolstoy, Extremal projectors for quantized Kac–Moody
superalgebras and some of their applications, in: Quantum Groups,
eds. H.D. Doebner and J.D. Hennig, Lecture Notes in Physics, Vol.
370 (Springer, Berlin, 1990) pp. 118–125,
http://dx.doi.org/10.1007/3-540-53503-9_45
[27] V.N. Tolstoy, Projection operator method for quantum groups,
in: Special Functions 2000, NATO Science Series II, Vol. 30
(Kluwer Acad. Publ., The Netherlands, 2001) pp. 457–488,
http://www.springer.com/mathematics/analysis/book/978-0-7923-7120-5
[28] A.I. Molev, Gelfand–Tsetlin basis for Yangians, Lett. Math.
Phys. 30, 53–60 (1994),
http://dx.doi.org/10.1007/BF00761422
[29] A. Klimyk and K. Schmüdgen, Quantum Groups and Their
Representations, Texts and Monographs in Physics
(Springer-Verlag, Berlin, 1997),
http://www.amazon.co.uk/Quantum-Groups-Representations-Monographs-Physics/dp/3540634525/
[30] K. Ueno, T. Takebayashi, and Y. Shibukawa, Construction of
Gel’fand–Zetlin basis for Uq(gl(N +
1)) modules, Publ. RIMS Kyoto Univ. 26, 667–679 (1990),
http://dx.doi.org/10.2977/prims/1195170852
[31] V.N. Tolstoy, Fortieth anniversary of extremal projector method
for Lie symmetries, in: Noncommutative Geometry and
Representation Theory in Mathematical Physics, eds. J. Fuchs,
J. Mickelsson, G. Rozenblioum, A. Stolin, and A. Westerberg,
Contemporary Mathematics, Vol. 391 (American Mathematical Society,
2005) p. 371–384 (2005),
http://www.ams.org/bookstore?fn=20&arg1=mathphys&ikey=CONM-391
[32] M.E. Taylor, Noncommutative Harmonic Analysis (American
Mathematical Society, Providence, 1986),
http://www.amazon.co.uk/Noncommutative-Harmonic-Analysis-Mathematical-Monographs/dp/0821815237/
[33] D.P. Zelobenko, Compact Lie Groups and Their
Representations (American Mathematical Society, Providence,
1973),
http://www.amazon.co.uk/Compact-Lie-Groups-Their-Representations/dp/0821815903/
[34] W. Fulton and J. Harris, Representation Theory: A First
Course (Springer-Verlag, New York, 1991),
http://www.amazon.co.uk/Representation-Theory-Course-Graduate-Mathematics/dp/3540974954/
[35] M. Kashiwara, On crystal bases of the q-analogue of
universal enveloping algebras, Duke Math. J. 63, 465–516
(1991),
http://dx.doi.org/10.1215/S0012-7094-91-06321-0
[36] G. Lusztig, Canonical bases arising from quantized enveloping
algebras, J. Am. Math. Soc. 2, 447–498 (1990),
http://dx.doi.org/10.1090/S0894-0347-1990-1035415-6
[37] J. Hong and H. Lee, Young tableaux and crystal B() for
finite simple Lie algebras, J. Algebra 320, 3680–3693
(2008),
http://dx.doi.org/10.1016/j.jalgebra.2008.06.008
[38] B. Leclerc and P. Toffin, A simple algorithm for computing the
global crystal basis of an irreducible Uq(sln)-module,
Int. J. Algebra Comput. 10, 191–208 (2000),
http://dx.doi.org/10.1142/S0218196700000042
[39] H. Lee, Nakajima monomials and crystals for special linear Lie
algebras, Nagoya Math. J. 188, 31–57 (2007),
http://projecteuclid.org/euclid.nmj/1197908743
[40] J. Paldus and C.R. Sarma, Clifford algebra unitary group
approach to many-electron correlation problem, J. Chem. Phys. 83,
5135–5152 (1985),
http://dx.doi.org/10.1063/1.449726
[41] F.A. Matsen, Spin-free quantum chemistry. XXIII. The
generator-state approach, Int. J. Quantum Chem. 32, 71–86
(1987),
http://dx.doi.org/10.1002/qua.560320108
[42] D.J. Rowe, Properties of overcomplete and nonorthogonal basis
vectors, J. Math. Phys. 10, 1774–1777 (1969),
http://dx.doi.org/10.1063/1.1665026
[43] P.O. Löwdin, Quantum theory of cohesive properties of solids,
Adv. Phys. 5, 1–172 (1956),
http://dx.doi.org/10.1080/00018735600101155
[44] R.W. Carter and G. Lusztig, On the modular representations of
the general linear and symmetric groups, Math. Z. 136,
193–242 (1974),
http://dx.doi.org/10.1007/BF01214125
[45] J.E. Humphreys, Introduction to Lie Algebras and
Representation Theory (Springer-Verlag, New York, 1972,
http://www.amazon.com/Introduction-Algebras-Representation-Graduate-Mathematics/dp/0387900535
[46] F.A. Matsen, Canonical generator states and their symmetry
adaptation, Int. J. Quant. Chem. 18, 43–56 (1984),
http://dx.doi.org/10.1002/qua.560260808
[47] K. Akin, D. Buchsbaum, and J. Weyman, Schur functors and Schur
complexes, Adv. Math. 44, 207–278 (1982),
http://dx.doi.org/10.1016/0001-8708(82)90039-1
[48] B.E. Sagan, The Symmetric Group, 2nd ed.
(Springer-Verlag, New York, 2001),
http://www.springer.com/mathematics/algebra/book/978-0-387-95067-9