In mathematical invariant theory, a perpetuant is informally an irreducible covariant of a form or infinite degree. More precisely, the dimension of the space of irreducible covariants of given degree and weight for a binary form stabilizes provided the degree of the form is larger than the weight of the covariant, and the elements of this space are called perpetuants. Perpetuants were introduced and named by Sylvester (1882, p.105). MacMahon (1884, 1885, 1894) and Stroh (1890) classified the perpetuants. Elliott (1907) describes the early history of perpetuants and gives an annotated bibliography.

MacMahon conjectured and Stroh proved that the dimension of the space of perpetuants of degree n>2 and weight w is the coefficient of x^w of

${\displaystyle {\frac {x^{2^{n-1}-1}}{(1-x^{2})(1-x^{3})\cdots (1-x^{n})}}}$

For n=1 there is just one perpetuant, of weight 0, and for n=2 the number is given by the coefficient of x^w of x²/(1-x²).

There are very few papers after about 1910 discussing perpetuants; (Littlewood 1944) is one of the few exceptions. (Kraft & Procesi 2020) exhibited an explicit base of the space of perpetuants.

References

• Cayley, Arthur (1884), "A Memoir on Seminvariants", American Journal of Mathematics, The Johns Hopkins University Press, 7 (1): 1–25, doi:10.2307/2369456, ISSN 0002-9327, JSTOR 2369456
• Elliott, Edwin Bailey (1895), "An introduction to the algebra of quantics", Nature, Oxford, Clarendon Press, 53 (1364): 147–148, Bibcode:1895Natur..53..147G, doi:10.1038/053147a0, S2CID 4036717, Reprinted by Chelsea Scientific Books 1964
• Elliott, Edwin Bailey (1907), "On Perpetuants and Contra-Perpetuants", Proc. London Math. Soc., 4 (1): 228–246, doi:10.1112/plms/s2-4.1.228
• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge University Press
• Kraft, Hanspeter; Procesi, Claudio (2020-03-11). "Perpetuants: A Lost Treasure". International Mathematics Research Notices. 2021 (5): 3597–3632. arXiv:1810.01131. doi:10.1093/imrn/rnaa032. ISSN 1073-7928.
• Littlewood, D. E. (1944), "Invariant theory, tensors and group characters", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 239 (807): 305–365, Bibcode:1944RSPTA.239..305L, doi:10.1098/rsta.1944.0001, ISSN 0080-4614, JSTOR 91389, MR 0010594
• MacMahon, P. A. (1884), "On Perpetuants", American Journal of Mathematics, The Johns Hopkins University Press, 7 (1): 26–46, doi:10.2307/2369457, ISSN 0002-9327, JSTOR 2369457
• MacMahon, P. A. (1885), "A Second Paper on Perpetuants", American Journal of Mathematics, The Johns Hopkins University Press, 7 (3): 259–263, doi:10.2307/2369271, ISSN 0002-9327, JSTOR 2369271
• MacMahon, P. A. (1894), "The Perpetuant Invariants of Binary Quantics", Proc. London Math. Soc., 26 (1): 262–284, doi:10.1112/plms/s1-26.1.262
• Stroh, E. (1888), "Ueber eine fundamentale Eigenschaft des Ueberschiebungs-processes und deren Verwerthung in der Theorie der binären Formen", Mathematische Annalen, Springer Berlin / Heidelberg, 33: 61–107, doi:10.1007/bf01444111, ISSN 0025-5831, S2CID 121256190
• Stroh, E. (1890), "Ueber die symbolische Darstellung der Grundsyzyganten einer binären Form sechster Ordnung und eine Erweiterung der Symbolik von Clebsch", Mathematische Annalen, Springer Berlin / Heidelberg, 36 (2): 262–303, doi:10.1007/BF01207843, S2CID 121641107
• Sylvester, James Joseph (1882), "On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order", American Journal of Mathematics, The Johns Hopkins University Press, 5 (1): 79–136, doi:10.2307/2369536, ISSN 0002-9327, JSTOR 2369536