Geodesic
ON THE LINEAR COMBINANTS OF A BINARY PENCIL
Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 481-498

Voir la notice de l'article provenant de la source Cambridge University Press

Let A, B denote binary forms of order d, and let 2r−1 = (A, B)2r−1 be the sequence of their linear combinants for . It is known that 1, 3 together determine the pencil {A + λ B}λ∈P1 and hence indirectly the higher combinants 2r−1. In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2r−1 from the knowledge of 1 and 3. The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧2Sd for the group SL2. We give an example for the group SL3 to show that such a result may hold for other categories of representations.
DOI : 10.1017/S0017089509005126
Mots-clés : 13A50, 22E70 
ABDESSELAM, ABDELMALEK; CHIPALKATTI, JAYDEEP. ON THE LINEAR COMBINANTS OF A BINARY PENCIL. Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 481-498. doi: 10.1017/S0017089509005126
@article{10_1017_S0017089509005126,
     author = {ABDESSELAM, ABDELMALEK and CHIPALKATTI, JAYDEEP},
     title = {ON {THE} {LINEAR} {COMBINANTS} {OF} {A} {BINARY} {PENCIL}},
     journal = {Glasgow mathematical journal},
     pages = {481--498},
     year = {2009},
     volume = {51},
     number = {3},
     doi = {10.1017/S0017089509005126},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089509005126/}
}
TY  - JOUR
AU  - ABDESSELAM, ABDELMALEK
AU  - CHIPALKATTI, JAYDEEP
TI  - ON THE LINEAR COMBINANTS OF A BINARY PENCIL
JO  - Glasgow mathematical journal
PY  - 2009
SP  - 481
EP  - 498
VL  - 51
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089509005126/
DO  - 10.1017/S0017089509005126
ID  - 10_1017_S0017089509005126
ER  - 
%0 Journal Article
%A ABDESSELAM, ABDELMALEK
%A CHIPALKATTI, JAYDEEP
%T ON THE LINEAR COMBINANTS OF A BINARY PENCIL
%J Glasgow mathematical journal
%D 2009
%P 481-498
%V 51
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089509005126/
%R 10.1017/S0017089509005126
%F 10_1017_S0017089509005126

[1] 1.Abdesselam, A. and Chipalkatti, J., The bipartite Brill–Gordan locus and angular momentum, Transform. Grp. 11 (3) (2006), 341–370. Google Scholar | DOI

[2] 2.Abdesselam, A. and Chipalkatti, J., On the Wronskian combinants of binary forms, J. Pure Appl. Algebra 210 (1) (2007), 43–61. Google Scholar | DOI

[3] 3.Abdesselam, A. and Chipalkatti, J., The higher transvectants are redundant. Preprint arXiv:0801.1533v1 [math.AG] (2008). Google Scholar

[4] 4.Carré, C. and Leclerc, B., Splitting the square of a Schur function into its symmetric and antisymmetric parts, J. Algebraic Combin. 4 (3) (1995), 201–231. Google Scholar | DOI

[5] 5.Chipalkatti, J., On the invariant theory of the Bézoutiant, Beiträge Algebra Geom. 47 (2) (2006), 397–417. Google Scholar

[6] 6.Clebsch, A., Theorie der Binaren Algebraischen Formen (Teubner, Leipzig, Germany, 1872). Google Scholar

[7] 7.Dolgachev, I., Lectures on invariant theory, London Mathematical Society Lecture Notes No. 296 (Cambridge University Press, Cambridge, 2003). Google Scholar | DOI

[8] 8.Fulton, W. and Harris, J., Representation theory: A first course, Graduate Texts in Mathematics. (Springer, New York, 1991). Google Scholar

[9] 9.Glenn, O., The theory of invariants (Ginn and Co., Boston, 1915). Google Scholar

[10] 10.Gordan, P., Ueber Combinanten, Math. Ann. 5 (1872), 95–122. Google Scholar | DOI

[11] 11.Grace, J. H. and Young, A., The algebra of invariants (Chelsea, New York, 1962). Google Scholar

[12] 12.Meulien, M., Sur les invariants des pinceaux de formes quintiques binaires, Ann. Inst. Fourier (Grenoble) 54 (2004), 21–51. Google Scholar | DOI

[13] 13.Newstead, P. E., Covariants of pencils of binary cubics, Proc. R. Soc. Edinburgh Sect. A, 91 (3-4) (1981–1982), 181–183. Google Scholar | DOI

[14] 14.Olver, P., Classical invariant theory, London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 1999). Google Scholar | DOI

[15] 15.Shenton, W., Linear combinants of systems of binary forms, with the syzygies of the second degree connecting them, Am. J. Math. 37 (3) (1915), 247–271. Google Scholar | DOI

[16] 16.Shepherd-Barron, N. I., The rationality of some moduli spaces of plane curves, Compositio Math. 67 (1988), 51–88. Google Scholar

[17] 17.Sturmfels, B., Algorithms in invariant theory, Texts and Monographs in Symbolic Computation (Springer, Vienna, 1993). Google Scholar | DOI

[18] 18.Wall, C. T. C., Pencils of binary quartics, Rend. Sem. Mat. Univ. Padova 99 (1998), 197–217. Google Scholar

Cité par Sources :