Symmetric functions and the phase problem in crystallography

Authors:
J. Buhler and Z. Reichstein

Journal:
Trans. Amer. Math. Soc. **357** (2005), 2353-2377

MSC (2000):
Primary 05E05, 13A50, 13P99, 20C10

Published electronically:
August 11, 2004

MathSciNet review:
2140442

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Abstract | References | Similar Articles | Additional Information

Abstract: The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with equal atoms in the unit cell, with small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.

**[BCP]**W. Bosma, J. Cannon, C. Playoust,*The Magma algebra system. I. The user language*, Computational algebra and number theory (London, 1993). J. Symbolic Comput.**24**(1997), no. 3-4, 235-265.**[Da]**John Dalbec,*Multisymmetric functions*, Beiträge Algebra Geom.**40**(1999), no. 1, 27–51. MR**1678567****[EM]**Shizuo Endô and Takehiko Miyata,*Quasi-permutation modules over finite groups*, J. Math. Soc. Japan**25**(1973), 397–421. MR**0318276****[F]**Daniel R. Farkas,*Reflection groups and multiplicative invariants*, Rocky Mountain J. Math.**16**(1986), no. 2, 215–222. MR**843049**, 10.1216/RMJ-1986-16-2-215**[Gia]**C. Giacovazzo,*Direct Phasing in Crystallography*, Oxford University Press, 1998.**[Ha]**H. Hauptman, The phase problem of x-ray crystallography, Rep. Prog. Phys. (1991), 1427-1454.**[HGXB]**H. Hauptman, D. Y. Guo, H. Xu, R. H. Blessing,*Algebraic Direct Methods for Few-Atom Structure Models*, Acta Crystallographica A**58**(2002), 361-369.**[HK]**Mowaffaq Hajja and Ming-chang Kang,*Twisted actions of symmetric groups*, J. Algebra**188**(1997), no. 2, 626–647. MR**1435378**, 10.1006/jabr.1996.6857**[KM]**Deepak Kapur and Klaus Madlener,*A completion procedure for computing a canonical basis for a 𝑘-subalgebra*, Computers and mathematics (Cambridge, MA, 1989) Springer, New York, 1989, pp. 1–11. MR**1005954****[L]**Nicole Marie Anne Lemire,*Reduction in the rationality problem for multiplicative invariant fields*, J. Algebra**238**(2001), no. 1, 51–81. MR**1822183**, 10.1006/jabr.2000.8652**[M]**Jörn Müller-Quade and Rainer Steinwandt,*Gröbner bases applied to finitely generated field extensions*, J. Symbolic Comput.**30**(2000), no. 4, 469–490. MR**1784753**, 10.1006/jsco.1999.0417**[Re]**Zinovy Reichstein,*SAGBI bases in rings of multiplicative invariants*, Comment. Math. Helv.**78**(2003), no. 1, 185–202. MR**1966757**, 10.1007/s000140300008**[RS]**Lorenzo Robbiano and Moss Sweedler,*Subalgebra bases*, Commutative algebra (Salvador, 1988) Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61–87. MR**1068324**, 10.1007/BFb0085537**[Ry]**Herbert John Ryser,*Combinatorial mathematics*, The Carus Mathematical Monographs, No. 14, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc., New York, 1963. MR**0150048****[Sh]**Igor R. Shafarevich,*Basic algebraic geometry. 1*, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR**1328833****[St1]**Bernd Sturmfels,*Algorithms in invariant theory*, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR**1255980****[St2]**Bernd Sturmfels,*Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR**1363949****[Sw]**Moss Sweedler,*Using Groebner bases to determine the algebraic and transcendental nature of field extensions: return of the killer tag variables*, Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993) Lecture Notes in Comput. Sci., vol. 673, Springer, Berlin, 1993, pp. 66–75. MR**1251970**, 10.1007/3-540-56686-4_34**[vW]**B. L. van der Waerden,*Algebra*, 8th edition, Springer-Verlag, 1971.

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Additional Information

**J. Buhler**

Affiliation:
Department of Mathematics, Reed College, Portland, Oregon 97202

Email:
jpb@reed.edu

**Z. Reichstein**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Email:
reichst@math.ubc.ca

DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03550-0

Keywords:
Crystallography,
structure factor,
phase problem,
symmetric function,
group action,
field of invariants,
SAGBI basis,
algorithmic computation,
multiplicative invariant,
rational invariant field

Received by editor(s):
January 2, 2003

Received by editor(s) in revised form:
October 15, 2003

Published electronically:
August 11, 2004

Additional Notes:
The second author was partially supported by NSF grant DMS-901675 and by an NSERC research grant

Article copyright:
© Copyright 2004
American Mathematical Society