The higher transvectants are redundant

Abdelmalek Abdesselam[1]; Jaydeep Chipalkatti[2]

  • [1] University of Virginia Department of Mathematics Kerchof Hall P. O. Box 400137 Charlottesville, VA 22904-4137 (USA)
  • [2] University of Manitoba Department of Mathematics 433 Machray Hall Winnipeg MB R3T 2N2 (Canada)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 5, page 1671-1713
  • ISSN: 0373-0956

Abstract

top
Let A , B denote generic binary forms, and let 𝔲 r = ( A , B ) r denote their r -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the { 𝔲 r } . As a consequence, we show that each of the higher transvectants { 𝔲 r : r ≥ 2 } is redundant in the sense that it can be completely recovered from 𝔲 0 and 𝔲 1 . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of S L 2 -representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.We give explicit computational examples for S L 3 , 𝔤 2 and 𝔖 5 to show that this result has possible analogues for other categories of representations.

How to cite

top

Abdesselam, Abdelmalek, and Chipalkatti, Jaydeep. "The higher transvectants are redundant." Annales de l’institut Fourier 59.5 (2009): 1671-1713. <http://eudml.org/doc/10438>.

@article{Abdesselam2009,
abstract = {Let $A,B$ denote generic binary forms, and let $\mathfrak\{u\}_r = (A,B)_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\lbrace \mathfrak\{u\}_r\rbrace $. As a consequence, we show that each of the higher transvectants $\lbrace \mathfrak\{u\}_r: r \ge 2\rbrace $ is redundant in the sense that it can be completely recovered from $\mathfrak\{u\}_0$ and $\mathfrak\{u\}_1$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $SL_2$-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.We give explicit computational examples for $SL_3, \mathfrak\{g\}_2$ and $\mathfrak\{S\}_5$ to show that this result has possible analogues for other categories of representations.},
affiliation = {University of Virginia Department of Mathematics Kerchof Hall P. O. Box 400137 Charlottesville, VA 22904-4137 (USA); University of Manitoba Department of Mathematics 433 Machray Hall Winnipeg MB R3T 2N2 (Canada)},
author = {Abdesselam, Abdelmalek, Chipalkatti, Jaydeep},
journal = {Annales de l’institut Fourier},
keywords = {Angular momentum in quantum mechanics; binary forms; Cauchy exact sequence; 9-j symbols; representations of $SL_2$; transvectants; invariant theory; representation theory},
language = {eng},
number = {5},
pages = {1671-1713},
publisher = {Association des Annales de l’institut Fourier},
title = {The higher transvectants are redundant},
url = {http://eudml.org/doc/10438},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Abdesselam, Abdelmalek
AU - Chipalkatti, Jaydeep
TI - The higher transvectants are redundant
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 1671
EP - 1713
AB - Let $A,B$ denote generic binary forms, and let $\mathfrak{u}_r = (A,B)_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\lbrace \mathfrak{u}_r\rbrace $. As a consequence, we show that each of the higher transvectants $\lbrace \mathfrak{u}_r: r \ge 2\rbrace $ is redundant in the sense that it can be completely recovered from $\mathfrak{u}_0$ and $\mathfrak{u}_1$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $SL_2$-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum.We give explicit computational examples for $SL_3, \mathfrak{g}_2$ and $\mathfrak{S}_5$ to show that this result has possible analogues for other categories of representations.
LA - eng
KW - Angular momentum in quantum mechanics; binary forms; Cauchy exact sequence; 9-j symbols; representations of $SL_2$; transvectants; invariant theory; representation theory
UR - http://eudml.org/doc/10438
ER -

References

top
  1. A. Abdesselam, The combinatorics of classical invariant theory revisited by modern physics, Slides of Feb 2007 talk at the Montreal CRM workshop “Combinatorial Problems Raised by Statistical Mechanics”. Available at http://people.virginia.edu/aa4cr/MontrealFeb07slides.pdf 
  2. A. Abdesselam, J. Chipalkatti, The bipartite Brill-Gordan locus and angular momentum, Transform. Groups 11 (2006), 341-370 Zbl1103.14030MR2264458
  3. A. Abdesselam, J. Chipalkatti, Brill-Gordan Loci, transvectants and an analogue of the Foulkes conjecture, Adv. Math 208 (2007), 491-520 Zbl1131.05005MR2304326
  4. K. Akin, D. Buchsbaum, J. Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982), 207-278 Zbl0497.15020MR658729
  5. S. J. Ališauskas, A. P. Jucys, Weight lowering operators and the multiplicity-free isoscalar factors for the group R 5 , J. Math. Phys. 12 (1971), 594-605 Zbl0214.28902MR286378
  6. L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics. Theory and Application, 8 (1981), Addison-Wesley Zbl0474.00023MR635121
  7. N. Bourbaki, Topological Vector Spaces, (1987), Springer-Verlag Zbl0622.46001MR910295
  8. J. Brunnemann, T. Thiemann, Simplification of the spectral analysis of the volume operator in loop quantum gravity, Class. Quant. Grav. 23 (2006), 1289-1346 Zbl1089.83013MR2205485
  9. P. J. Brussaard, Clebsch-Gordan- of Wigner coefficienten, Ned. Tijdschr. Natuurk. 33 (1967), 202-222 
  10. J. Carter, D. Flath, M. Saito, The Classical and Quantum 6j-Symbols, (1995), Princeton University Press Zbl0851.17001MR1366832
  11. A. Cayley, On linear transformations, (1889), Cambridge University Press Zbl01.0277.03
  12. J. Chipalkatti, On the invariant theory of the Bezoutiant, Beiträge Alg. Geom. 47 (2006), 397-417 Zbl1112.13009MR2307911
  13. A. Clebsch, Theorie der Binaren Algebraischen Formen, (1872), Teubner, Leipzig Zbl02.0058.02
  14. E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra, (1935), Cambridge University Press Zbl0014.04605
  15. I. Dolgachev, Lectures on Invariant Theory, (2003), Cambridge University Press Zbl1023.13006MR2004511
  16. A. R. Edmonds, Angular Momentum in Quantum Mechanics, (1957), Princeton University Press Zbl0079.42204MR95700
  17. D. Flath, The Clebsch-Gordan formulas, Enseign. Math. (2) 29 (1983), 339-346 Zbl0524.17003MR719316
  18. W. Fulton, Young Tableaux, (1957), Cambridge University Press Zbl0878.14034MR1464693
  19. W. Fulton, J. Harris, Representation Theory, A First Course, (1991), Springer–Verlag Zbl0744.22001MR1153249
  20. O. Glenn, The Theory of Invariants, (1915), Ginn and Co., Boston Zbl45.0240.01
  21. L. Goldberg, Catalan numbers and branched coverings by the Riemann sphere, Adv. Math. 85 (1991), 129-144 Zbl0732.14013MR1093002
  22. P. Gordan, Die Resultante Binärer Formen, Rend. Circ. Matem. Palermo XXII (1906), 161-196 Zbl37.0192.08
  23. J. H. Grace, A. Young, The Algebra of Invariants, 1903, (1962), Reprinted by Chelsea Publishing Co., New York Zbl34.0114.01
  24. J. Harris, Algebraic Geometry, A First Course, (1992), Springer–Verlag Zbl0779.14001MR1182558
  25. R. Hartshorne, Algebraic Geometry, (1977), Springer-Verlag Zbl0367.14001MR463157
  26. J.-S. Huang, C.-B. Zhu, Weyl’s construction and tensor power decomposition for G 2 , Proc. Amer. Math. Soc. 127 (1999), 925-934 Zbl0912.22006MR1469412
  27. H. A. Jahn, J. Hope, Symmetry properties of the Wigner 9j symbol, Phys. Rev. 93 (1954), 318-321 Zbl0055.43704
  28. J. Van der Jeugt, N. Pitre Sangita, K. Srinivasa Rao, Multiple hypergeometric functions and 9-j coefficients, J. Phys A: Math. Gen. 27 (1994), 5251-5264 Zbl0847.33009MR1295355
  29. A. P. Jucys, A. A. Bandzaitis, Angular Momentum in Quantum Physics, (1977), Vilnius: Mokslas 
  30. A. A. Kirillov, Elements of the Theory of Representations, (1976), Springer-Verlag Zbl0342.22001MR412321
  31. J. P. S. Kung, G.-C. Rota, The invariant theory of binary forms, Bulletin of the A.M.S. 10 (1984), 27-85 Zbl0577.15020MR722856
  32. D. E. Littlewood, Invariant theory, tensors and group characters, Philos. Trans. Roy. Soc. London. Ser. A 239 (1944), 305-365 Zbl0060.04402MR10594
  33. I. G. MacDonald, Symmetric Functions and Hall polynomials, (1995), Oxford University Press Zbl0824.05059MR1354144
  34. D. Mumford, Lectures on Curves on an Algebraic Surface, (1966), Princeton University Press Zbl0187.42701MR209285
  35. P. Olver, Classical Invariant Theory, (1999), Cambridge University Press Zbl0971.13004MR1694364
  36. G. Racah, Theory of complex spectra II, Phys. Rev. 62 (1942), 438-462 
  37. H. Rosengren, On the triple sum formula for Wigner 9j-symbols, J. Math. Phys. 39 (1998), 6730-6744 Zbl0935.81036MR1660852
  38. H. Rosengren, Another proof of the triple sum formula for Wigner 9j-symbols, J. Math. Phys. 40 (1999), 6689-6691 Zbl0967.81032MR1725880
  39. G. Salmon, Higher Algebra, (1965), Reprinted by Chelsea Publishing Co., New York 
  40. T. A. Springer, Invariant Theory, (1977), Springer-Verlag Zbl0346.20020MR447428
  41. E. Stroh, Entwicklung der Grundsyzyganten der binären Form fünfter Ordnung, Math. Ann. 34 (1889), 354-370 Zbl21.0108.01MR1510582
  42. B. Sturmfels, Algorithms in Invariant Theory, (1993), Springer-Verlag Zbl0802.13002MR1255980
  43. E. Wigner, Group Theory and Its Application to the Quantum Theory of Atomic Spectra, (1959), Academic Press Zbl0085.37905MR106711

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.