Multiple antisymmetry
From Wikipedia, the free encyclopedia
In crystallography and materials science multiple antisymmetry is the concept of a substance possessing two or more different types of antisymmetry simultaneously. Multiple antisymmetry has applications in the fields of magnetic structures, ferroelectricity, and the physical properties of crystals.
Antisymmetry is a symmetry operation which reverses an object to its opposite.[1] A more precise definition is "operations of antisymmetry transform objects possessing two possible values of a given property from one value to the other."[2]
An antisymmetry can be any two-valued operation which, if executed twice, returns the object to its original state. In the 1930s the first use of an antisymmetry operation was dichromatic symmetry or black-white symmetry.[3][4][5] In 1951 Landau and Lifshitz reinterpreted black and white colours to correspond to time-reversal symmetry.[6]
A wide variety of antisymmetries are now recognized, with three new antisymmetries having been discovered since 2010.[7]
| Antisymmetry | Transformation | Symbol[7] | Refs. |
|---|---|---|---|
| Black-white (2-colour) |  |  | [8][9] |
| Space inversion |  |  | [10] |
| Time-reversal |  |  or | [6][11] |
| Charge-reversal |  |  or | [12] |
| Magnetic moment |  |  or | [13] |
| Rotation-reversal |  |  | [14] |
| Distortion-reversal |  |  | [15][16][17] |
| Wedge reversion |  |  | [18] |
A symmetry operation which interchanges three or more colours is not an antisymmetry but rather a polychromatic or colour symmetry.
Multiple antisymmetry
If two different antisymmetry operations are defined (for example
Where there are 
| Group | | Symbol[20] | Number | Refs. |
|---|---|---|---|---|
| Point groups in 3D | 0 |  | 32 | [21] |
| 1 |  | 122 | [4][8][9] | |
| 2 |  | 624 | [19][22] | |
| 3 |  | 4362 | [23] | |
| 4 |  | 42244 | [23] | |
| Space groups in 3D | 0 |  | 230 | [24] |
| 1 |  | 1651 | [25][26] | |
| 2 |  | 17803 | [19][22] | |
| 3 |  | 287574 | [15] | |
| 4 |  | 6880800 | [23] |
Applications
Simultaneously with the theoretical investigations of A. M. Zamorzaev, practical applications of double antisymmetry were being pursued by other Russian researchers such as B. A. Tavger, B. K. Vainshtein and L. A. Shuvalov.
In 1962 Shuvalov and N. V. Belov used two different types of antisymmetry to describe the combination of ferromagnetic and ferroelectric properties of crystals.[27][28] This field is now termed multiferroics, and has been an intensive area of research since 2000.[29] Magnetochromism is a related field in which magnetic moment and black-white antisymmetries are combined.[30]
Space-time reversal (inverting both space and time coordinates, signified by 
The combination of space reversal and magnetic reversal was studied by Vojtěch Kopský in 2006.[33]
Rotation-reversal space groups have been used in the classification of tilted octahedra perovskites.[34][19]
History
The first suggestion of the concept of multiple antisymmetry[Note 1] was in a 1944 paper by Alexei Vasilievich Shubnikov.[35] The field was developed by Alexander Mihailovich Zamorzaev and his research group at Moldova State University.[36]
1950s to 1970s
In 1957 Zamorzaev and Sokolov introduced the concept of more than one kind of two-valued antisymmetry operation (multiple antisymmetry) and derived the crystallographic (finite) point groups with one or more types of antisymmetry.[37] In 1958 Zamorzaev derived the space groups with two types of antisymmetry.[38] From 1960–63 Zamorzaev and his co-workers derived the two-dimensional (planar) groups with two or three types of antisymmetry,[39][40] the point groups with three types of antisymmetry,[41] and the layer groups with two types of antisymmetry.[42] In 1963 Shuvalov derived the limit groups of double antisymmetry.[43]
From 1964–1965 Zamorzaev derived the similarity groups with two or three types of antisymmetry,[44] the layer groups with two to five types of antisymmetry,[45] the groups of borders and ribbons[Note 2] with two or three types of antisymmetry,[46][47] the space groups with three to six types of antisymmetry,[48] one- and two-sided rosettes, and finite borders and ribbons with two to four types of antisymmetry,[49] and rod groups with two to four types of antisymmetry.[50]
In 1967 Zamorzaev derived the two-dimensional groups with colour antisymmetry of two different types,[51] and introduced the concept of quasisymmetry (P-symmetry).[Note 3] P-symmetry is a type of generalized symmetry where P represents a permutation group acting on a set of indices.[52][53] In 1980 Alexandru Lungu, one of Zamorzaev's students, introduced Q-symmetry. Q-symmetry is a type of generalized symmetry which involves transformations on weighted figures and is based on subgroups of semidirect products. Lungu went on to introduce Wp and Wq symmetries based on wreath products.[54] In 1982 Vojtěch Kopský showed that Q- and Wq-symmetries can be reduced to P- and Wp-symmetries.[55][56] A few papers on multiple antisymmetry were also published by non-Russian authors.[57][58]
Reviews of the field were published by Koptsik,[59] Zamorzaev,[60][23] and Shubnikov and Koptsik.[61] Books summarizing the work of the Zamorzaev school were published in the 1970s and 1980s in Russian.[62][63][64]
At the end of the 1970s the Russian research entered a period of diminishing returns, as the complexity of the results increased, but the usefulness of the results decreased. In practice, a single type of antisymmetry was a useful concept in explaining experimental results in a wide variety of physical fields. Two types of antisymmetry were rarely required to explain a physical result, and three or more types of antisymmetry were not required in practice.
1980s to 1990s
From the mid-1980s to the mid-2000s Slavik Vlado Jablan worked in the field of multiple antisymmetry. In 1986 Jablan defined a method of generating antisymmetry groups using the new concept of "antisymmetric characteristic".[65] Vojtěch Kopský[66] criticised the paper in Mathematical Reviews: "The latter concept is not clearly defined and theorems are given without proof, which makes it hard to follow the logic of the paper."[67]
Jablan followed up with a long series of papers cataloguing multiple antisymmetry groups using his new antisymmetric characteristic method.[68] These papers were critically reviewed in Mathematical Reviews.[69]
Where initially the major workers in the field (Zamorzaev, Palistrant, Jablan) had papers published in major journals such Acta Crystallographica and Zeitschrift für Kristallographie, their later papers were mostly published in minor journals. By 1992, in a review of one of Zamorzaev and Palistrant's papers,[70] Vojtěch Kopský (one of the two editors of the International Tables for Crystallography Volume E: Subperiodic groups[71]) commented: "The text is in narrative style and makes the impression that its aim is to promote the subject rather than to develop a consistent mathematical scheme. As usual, there are statements about the importance of the subject in physics, and this time there is a claim of contributions to the advance of crystallography in arbitrary dimensions. ... In the reviewer's opinion, for what it is worth, the whole subject is a dead branch of symmetry theory."[72]
2000s to 2020s
From the mid-1980s Daniel B. Litvin published extensively on magnetic space groups[32][73] and later worked on multiple antisymmetry.[22]
In the 2010s a new group of researchers entered the field including Ismaila Dabo, Venkatraman Gopalan, Mantao Huang, Vincent S. Liu, Jason M. Munro, Brian K. VanLeeuwen, and Haricharan Padmanabhan. They built on Litvin's work[74][75] and published papers on new types of antisymmetries, such as rotation-reversal,[14][19] distortion-reversal.[15][16] and wedge reversion.[18][76]
Notes
- ↑ Multiple antisymmetry was referred to as "generalized antisymmetry" and "different kinds of antisymmetry" by the early Russian authors.
- ↑ A border is a two-dimensional figure infinite in one direction (or a 2-sided line); a ribbon is a figure in a layer infinite in one direction (a 2-sided border, or a 4-sided line).
- ↑ In P-symmetry it is assumed that a geometric figure π with isometry group G is coloured so that each element of G effects a unique permutation of the colours.
References
- ↑ Atoji, Masao (1 March 1965). "Graphical Representations of Magnetic Space Groups". American Journal of Physics. 33 (3): 212–219. doi:10.1119/1.1971375.
- ↑ Mackay, A. L. (10 September 1957). "Extensions of space-group theory". Acta Crystallographica. 10 (9): 543–548. doi:10.1107/S0365110X57001966.
- ↑ Weber, L. (1929). "Die Symmetrie homogener ebener Punktsysteme" [The symmetry of homogeneous plane point systems]. Zeitschrift für Kristallographie (in German). 70: 309–327. doi:10.1524/zkri.1929.70.1.309.
- 1 2 Heesch, H. (1930). "Über die vierdimensionalen Gruppen des dreidimensionalen Raumes" [On the four-dimensional groups of three-dimensional space]. Zeitschrift für Kristallographie (in German). 73: 325–345. doi:10.1524/zkri.1930.73.1.325.
- ↑ Woods, H. J. (1936). "The geometric basis of pattern design part IV: counterchange symmetry in plane patterns". Journal of the Textile Institute, Transactions. 27: T305–320. doi:10.1080/19447023608661695.
- 1 2 Landau, L. D.; Lifshitz, E. M. (1951). Course of theoretical physics (in Russian). Vol. 5. Statistical physics (first ed.). Moscow: Nauka.
- 1 2 3 4 Padmanabhan, H., Munro, J. M., Dabo, I and Gopalan, V. (2020). Antisymmetry: fundamentals and applications, Annual Review of Materials Research, 50, 255–281, doi:10.1146/annurev-matsci-100219-101404
- 1 2 Shubnikov, A. V. (1951). Simmetriya i antisimmetriya konechnykh figur [Symmetry and antisymmetry of finite figures] (in Russian). Moscow: Izv. Akad. Nauk SSSR.
- 1 2 Shubnikov, A. V. (1964). "Symmetry and antisymmetry of finite figures". In Holser, William T. (ed.). Colored symmetry. New York, London: Pergamon Press. pp. xv-172. Retrieved 18 August 2025.
- ↑ Aroyo, Mois I., ed. (2016). International tables for crystallography. Vol. A. Dordrecht, Holland ; Boston, U.S.A. : Hingham, MA: D. Reidel Pub. Co. ; Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers Group. p. 13. doi:10.1107/97809553602060000114. ISBN 978-0-470-97423-0.
- ↑ Shubnikov, A. V. (1960). "Time reversal as an operation of antisymmetry". Soviet Physics Crystallography. 5 (2): 309–314.
- ↑ Geru, Ion I. (2019). Time-Reversal Symmetry: Seven Time-Reversal Operators for Spin Containing Systems. Cham: Springer International Publishing AG. doi:10.1007/978-3-030-01210-6. ISBN 978-3-030-01209-0.
- ↑ Tavger, B. A.; Zaitsev, V. M. (1956). "Magnetic symmetry of crystals" (PDF). Journal of Experimental and Theoretical Physics (Soviet Physics JETP). 3 (3): 430–436. Retrieved 18 August 2025.
- 1 2 Gopalan, Venkatraman; Litvin, Daniel B. (May 2011). "Rotation-reversal symmetries in crystals and handed structures". Nature Materials. 10 (5): 376–381. arXiv:1007.3544. doi:10.1038/nmat2987.
- 1 2 3 VanLeeuwen, Brian K.; Gopalan, Venkatraman (17 November 2015). "The antisymmetry of distortions". Nature Communications. 6 (8818). doi:10.1038/ncomms9818. PMC 4660051.
- 1 2 VanLeeuwen, Brian Kevin (2015). The symmetry and antisymmetry of distortions (PhD thesis). The Pennsylvania State University. Retrieved 22 August 2025.
- ↑ Munro, Jason M.; Akamatsu, Hirofumi; Padmanabhan, Haricharan; Liu, Vincent S.; Shi, Yin; Chen, Long-Qing; VanLeeuwen, Brian K.; Dabo, Ismaila; Gopalan, Venkatraman (6 August 2018). "Discovering minimum energy pathways via distortion symmetry groups". Physical Review B. 98 (8). arXiv:1804.06798. doi:10.1103/PhysRevB.98.085107.
- 1 2 Gopalan, Venkatraman (1 May 2020). "Wedge reversion antisymmetry and 41 types of physical quantities in arbitrary dimensions". Acta Crystallographica Section A: Foundations and Advances. 76 (3): 318–327. doi:10.1107/S205327332000217X. PMC 7233017.
- 1 2 3 4 5 VanLeeuwen, Brian K.; Gopalan, Venkatraman; Litvin, Daniel B. (1 January 2014). "Double antisymmetry and the rotation-reversal space groups". Acta Crystallographica Section A: Foundations and Advances. 70 (1): 24–38. doi:10.1107/S2053273313023176.
- ↑ Bohm, J.; Dornberger-Schiff, K. (1 December 1966). "The nomenclature of crystallographic symmetry groups". Acta Crystallographica. 21 (6): 1004–1007. doi:10.1107/S0365110X66004389.
- ↑ Frankenheim, M. L. (1826). "Crystallonomische Aufsätze" [Crystallographic essays]. Isis (Jena) (in German). 19: 497–515, 542–565. Retrieved 23 August 2025.
- 1 2 3 Huang, Mantao; VanLeeuwen, Brian K.; Litvin, Daniel B.; Gopalan, Venkatraman (1 July 2014). "Crystallographic data of double antisymmetry space groups". Acta Crystallographica Section A: Foundations and Advances. 70 (4): 373–381. doi:10.1107/s2053273314006871.
- 1 2 3 4 Zamorzaev, A. M. (1988). "Generalized antisymmetry". Computers and Mathematics with Applications. 16 (5–8): 555–562. doi:10.1016/0898-1221(88)90245-3.
- ↑ Burckhardt, J. J. (1967). "Zur Geschichte der Entdeckung der 230 Raumgruppen" [On the history of the discovery of the 230 space groups]. Archive for History of Exact Sciences (in German). 4 (3): 235–246. doi:10.1007/BF00412962.
- ↑ Zamorzaev, A. M. (1957). "Generalization of Fedorov groups". Soviet Physics Crystallography. 2 (1): 10–15. Retrieved 24 August 2025.
- ↑ Belov, N. V.; Neronova, N. N.; Smirnova, T. S. (1957). "Shubnikov groups". Soviet Physics Crystallography. 2 (3): 311–322.
- ↑ Shuvalov, L. A.; Belov, N. V. (1962). "The symmetry of crystals in which ferromagnetic and ferroelectric properties appear simultaneously". Soviet Physics Crystallography. 7 (2): 150–151.
- ↑ Shuvalov, L. A. (1963a). "Antisymmetry and its concrete modifications". Soviet Physics Crystallography. 7 (3): 418–423.
- ↑ Spaldin, Nicola A. (2020). "Multiferroics beyond electric-field control of magnetism". Proc. R. Soc. A. 476 (2233): 0542. arXiv:1908.08352. Bibcode:2020RSPSA.47690542S. doi:10.1098/rspa.2019.0542. PMC 7016559. PMID 32082059.
- ↑ Xu, X. S.; Brinzari, T. V.; Lee, S.; Chu, Y. H.; Martin, L. W.; Kumar, A.; McGill, S.; Rai, R. C.; Ramesh, R.; Gopalan, V.; Cheong, S. W.; Musfeldt, J. L. (21 April 2009). "Optical properties and magnetochromism in multiferroic BiFeO3". Physical Review B. 79 (13) 134425. doi:10.1103/PhysRevB.79.134425.
- ↑ Sirotin, Yu. I.; Shaskolskaya, M. P. (1982). Fundamentals Of Crystal Physics. Moscow: Mir Publishers. pp. 476–479. Retrieved 19 September 2025.
- 1 2 Litvin, D. B. (1 January 1986). "Ferroelectric space groups". Acta Crystallographica Section A: Foundations of Crystallography. 42 (1): 44–47. doi:10.1107/S0108767386099920.
- ↑ Kopský, Vojtěch (1 January 2006). "Opechowski's magic relations". Zeitschrift für Kristallographie - Crystalline Materials. 221 (1): 51–62. doi:10.1524/zkri.2006.221.1.51.
- ↑ Glazer, A.M. (May 2011). "A brief history of tilts". Phase Transitions. 84 (5–6): 405–420. doi:10.1080/01411594.2010.544732.
- ↑ Shubnikov, A. V. (1945). "Recent developments in the science of symmetry and its applications". General Assembly of the Academy of Sciences of the USSR on October 14-17, 1944 (in Russian). Moscow: Publishing House of Sciences of the USSR. pp. 212–227. Retrieved 19 August 2025.
- ↑ Palistrant, A. F. (2002). "Professor A. M. Zamorzaev (1927-1997) and Kishinev school of discrete geometry". VisMath. 4 (1). Retrieved 18 August 2025.
- ↑ Zamorzaev, A. M.; Sokolov, E. I. (1957). "Symmetry and various kinds of antisymmetry of finite bodies". Soviet Physics Crystallography. 2 (1): 5–9. Retrieved 18 August 2025.
- ↑ Zamorzaev, A. M. (1958). "Derivation of new Shubnikov groups". Soviet Physics Crystallography. 3 (4): 401–406.
- ↑ Zamorzaev, A. M.; Palistrant, A. F. (1960). "The two-dimensional Shubnikov groups". Soviet Physics Crystallography. 5 (4): 497–503.
- ↑ Zamorzaev, A. M.; Palistrant, A. F. (1961). "Mosaics for 167 two-dimensional Shubnikov groups (three lowest orders)". Soviet Physics Crystallography. 6 (2): 127–140.
- ↑ Galyarskii, E. I.; Zamorzaev, A. M. (1963). "Point symmetry groups and a different kind of antisymmetry". Soviet Physics Crystallography. 8 (1): 68–75.
- ↑ Palistrant, A. F.; Zamorzaev, A. M. (1963). "The symmetry and antisymmetry groups of layers". Soviet Physics Crystallography. 8 (2): 120–126.
- ↑ Shuvalov, L. A. (1963b). "Limit groups of double antisymmetry". Soviet Physics Crystallography. 7 (6): 669–672.
- ↑ Galyarskii, E. I.; Zamorzaev, A. M. (1964). "Similarity symmetric and antisymmetric groups". Soviet Physics Crystallography. 8 (5): 553–558.
- ↑ Palistrant, A. F. (1964). "Groups of layer symmetry and antisymmetry of different types". Soviet Physics Crystallography. 8 (5): 624–626.
- ↑ Palistrant, A. F.; Zamorzaev, A. M. (1964). "Groups of symmetry and of different types of antisymmetry of borders and ribbons". Soviet Physics Crystallography. 9 (2): 123–128.
- ↑ Neronova, N. N.; Belov, N. V. (1961). "A single scheme for the classical and black-and-white crystallographic symmetry groups". Soviet Physics Crystallography. 6 (1): 1–9.
- ↑ Zamorzaev, A. M.; Palistrant, A. F. (1965). "The number of generalized Shubnikov space groups". Soviet Physics Crystallography. 9 (6): 660–664.
- ↑ Palistrant, A. F. (1965). "Planar point groups of symmetry and different types of antisymmetry". Soviet Physics Crystallography. 10 (1): 1–5.
- ↑ Galyarskii, E. I.; Zamorzaev, A. M. (1965). "A complete derivation of crystallographic stem groups of symmetry and different types of antisymmetry". Soviet Physics Crystallography. 10 (2): 109–115.
- ↑ Palistrant, A. F. (1967). "Two-dimensional groups of color symmetry and color antisymmetry of different kinds:". Soviet Physics Crystallography. 11 (5): 609–613.
- ↑ Zamorzaev, A. M. (1967). "Quasisymmetry (P-symmetry)". Soviet Physics Crystallography. 12 (5): 717–722.
- ↑ Krishnamurty, T. S. G.; Prasad, L. S. R. K.; Rao, K. R. Mohana (May 1978). "On quasisymmetry (P-symmetry) groups". Journal of Physics A: Mathematical and General. 11 (5): 805–811. doi:10.1088/0305-4470/11/5/010.
- ↑ Lungu, A. P. (1968). "Derivation of Q-symmetry (P-symmetry) groups". Soviet Physics Crystallography. 25 (5): 602–604.
- ↑ Kopský, V. (January 1982a). "An algebraic study of contemporary symmetry concepts — Colour groups and related symmetries". Czechoslovak Journal of Physics. 32 (1): 3–18. doi:10.1007/bf01597539.
- ↑ Kopský, Vojtěch (August 1982b). "Colour symmetry polemics". Physica A: Statistical Mechanics and its Applications. 114 (1–3): 581–587. doi:10.1016/0378-4371(82)90353-3.
- ↑ Wondratschek, Hans; Niggli, Alfred (May 1961). "Eine Verallgemeinerung der Punktgruppen II. Die mehrfachen Kryptosymmetrien" [A generalization of point groups II. The multiple cryptosymmetries]. Zeitschrift für Kristallographie (in German). 115 (1–2): 1–20. doi:10.1524/zkri.1961.115.1-2.1.
- ↑ Sivardière, Jean (1971). "Groupes cristallographiques à simple, double et multiple antisymétrie" [Single, double and multiple antisymmetry crystallographic groups]. Bulletin de la Société française de Minéralogie et de Cristallographie (in French). 94 (1): 30–37. doi:10.3406/bulmi.1971.6547.
- ↑ Koptsik, V. A. (1968). "A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the last 50 years". Soviet Physics Crystallography. 12 (5): 667–683.
- ↑ Zamorzaev, A. M.; Palistrant, A. F. (1980). "Antisymmetry, its generalizations and geometrical applications". Zeitschrift für Kristallographie. 151: 231–248. doi:10.1524/zkri.1980.151.3-4.231.
- ↑ Shubnikov, A. V.; Kopt︠s︡ik, V. A. (1974). "Groups of generalized symmetry, antisymmetry and colored symmetry". In Harker, David (ed.). Symmetry in science and art. New York: Plenum Press. pp. 263–306. ISBN 0306307596. Retrieved 22 August 2025.
- ↑ Zamorzaev, A. M. (1976). Theory of Simple and Multiple Antisymmetry (in Russian). Kishinev: Shtiintsa.
- ↑ Zamorzaev, A. M.; Galyarsky, E. I.; Palistrant, A. F. (1978). Colour Symmetry, its Generalizations and Applications (in Russian). Kishinev: Shtiintsa.
- ↑ Zamorzaev, A. M.; Karpova, Yu. S.; Galyarsky, E. I.; Palistrant, A. F. (1986). P-symmetry and its Further Development (in Russian). Kishinev: Shtiintsa.
- ↑ Jablan, S. V. (1 July 1986). "A new method of generating plane groups of simple and multiple antisymmetry". Acta Crystallographica Section A: Foundations of Crystallography. 42 (4): 209–212. doi:10.1107/s0108767386099403.
- ↑ Litvin, D. B.; Janovec, V. (1 September 2016). "Vojtech Jaroslav Kopský (1936–2016)". Acta Crystallographica Section A: Foundations and Advances. 72 (5): 586–587. doi:10.1107/S2053273316011852.
- ↑ Kopský, Vojtěch. "A new method of generating plane groups of simple and multiple antisymmetry". MathSciNet (review). American Mathematical Society. MR 0848383. Retrieved 27 August 2025.
- ↑
- Jablan, S. V. (1 May 1987a). "A new method of deriving and cataloguing simple and multiple antisymmetry
space groups". Acta Crystallographica Section A: Foundations of Crystallography. 43 (3): 326–337. doi:10.1107/s010876738709932x.
- Jablan, Slavik V. (1987b). "Simple and multiple antisymmetry point groups
". Matematički Vesnik. 39 (3). Serbian Academy of Sciences and Arts: 301–308. Retrieved 28 August 2025.
- Jablan, Slavik V. (1990a). "Point groups of simple and multiple antihomology
". Zbornik radova Filozofskog fakulteta u Nišu. Serija Matematika. 4. University of Niš: 35–43. JSTOR 43999882.
- Jablan, Slavik (1990b). "Algebra of Antisymmetric Characteristics". Publications de l'Institut Mathématique (Beograd). 47 (61): 39–55. Retrieved 28 August 2025.
- Jablan, Slavik V. (1991). "Simple and multiple antisymmetry colorings of rosettes". Matematički Vesnik. 43 (1–2): 87–90. Retrieved 28 August 2025.
- Palistrant, A. F.; Jablan, S.V. (1991). "Enantiomorphism of three-dimensional space and line multiple antisymmetry groups" (PDF). Publications de l'Institut Mathématique (Beograd). 49 (63): 51–60. Retrieved 28 August 2025.
- Jablan, S. V. (1 January 1993). "Pawley multiple antisymmetry three-dimensional space groups
". Acta Crystallographica Section A: Foundations of Crystallography. 49 (1): 127–132. doi:10.1107/s0108767392005464.
- Jablan, Slavik V. (1996). "Pawley multiple antisymmetry three-dimensional space groups
- Jablan, S. V.; Radović, Ljiljana (2002). "Computational derivation of antisymmetry and colored symmetry groups". Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica (3): 87–96. Retrieved 28 August 2025.
- Jablan, S. V. (1 May 1987a). "A new method of deriving and cataloguing simple and multiple antisymmetry
- ↑
- Schwarzenberger, R. L. E. "Simple and multiple antisymmetry point groups
Can be understood only by reference to notation introduced by the author in previous papers
- Kopský, Vojtěch. "Point groups of simple and multiple antisymmetry
A method is used to go heavily into derivation, systematization and catalogization of nearly self-evident constructions
- Schwarzenberger, R. L. E. "Algebra of Antisymmetric Characteristics" (review). MR 1103527. Retrieved 28 August 2025.
Very difficult to follow given the author's insistence on giving examples rather than precise definitions
- Kopský, Vojtěch. "Simple and multiple antisymmetry colorings of rosettes" (review). MR 1210264. Retrieved 28 August 2025.
The solution of the title problem is a result of simple combinatorics. This is obscured rather than clarified by reference to the so-called method of «antisymmetry characteristics»
- Schwarzenberger, R. L. E. "Enantiomorphism of three-dimensional space and line multiple antisymmetry groups" (review). MR 1127378. Retrieved 28 August 2025.
An algorithm is explained, and numerical results obtained, but—as in previous papers—with no detailed proof
- Kopský, Vojtěch. "Pawley multiple antisymmetry three-dimensional space groups
The presentation is given in the jargon of antisymmetry investigators, behind which is hidden the fact that this type of work amounts to the investigation of subgroups and the construction of subdirect products
- Rossetti, Juan Pablo. "Pawley multiple antisymmetry three-dimensional space groups
It is difficult to understand the paper because the author assumes the reader's familiarity with the previous work of himself, Palistrant and Zamorzaev and some of these references are not easily available
- Szczepański, Andrzej. "Computational derivation of antisymmetry and colored symmetry groups" (review). MR 1991019. Retrieved 28 August 2025.
They use programs written in Mathematica 4. The article's presentation is not mathematical. Its theorems are stated without proof
- Schwarzenberger, R. L. E. "Simple and multiple antisymmetry point groups
- ↑ Zamorzaev, A. M.; Palistrant, A. F. (1992). "Space groups of generalized antisymmetry and their geometric applications". Space symmetry groups. Problems of Modern Crystallography (in Russian). Moscow: Nauka. pp. 100–112. ISBN 5-02-000223-2.
- ↑ Kopský, V.; Litvin, D. B., eds. (2010). International Tables for Crystallography. Vol. E: Subperiodic groups (Second online ed.). doi:10.1107/97809553602060000109. ISBN 978-0-470-68672-0.
- ↑ Kopský, Vojtěch (1992). "Space groups of generalized antisymmetry and their geometric applications". MathSciNet (review). American Mathematical Society. MR 1223111. Retrieved 27 August 2025.
- ↑
- Litvin, D. B.; Kotzev, J. N.; Birman, J. L. (15 December 1982). "Physical applications of crystallographic color groups: Landau theory of phase transitions". Physical Review B. 26 (12): 6947–6970. doi:10.1103/physrevb.26.6947.
- Litvin, Daniel B. (1 September 1999). "Magnetic subperiodic groups". Acta Crystallographica Section A: Foundations of Crystallography. 55 (5): 963–964. doi:10.1107/S0108767399003487.
- Litvin, Daniel B. (April 2013). Magnetic Group Tables: 1-, 2- and 3-dimensional magnetic subperiodic groups and magnetic space groups. International Union of Crystallography. doi:10.1107/9780955360220001. ISBN 978-0-9553602-2-0.
- ↑ Liu, Vincent S.; VanLeeuwen, Brian K.; Munro, Jason M.; Padmanabhan, Haricharan; Dabo, Ismaila; Gopalan, Venkatraman; Litvin, Daniel B. (1 July 2018). "Spatio-temporal symmetry – crystallographic point groups with time translations and time inversion". Acta Crystallographica Section A: Foundations and Advances. 74 (4): 399–402. doi:10.1107/S2053273318004667.
- ↑ Gopalan, Venkatraman (1 July 2021). "Relativistic spacetime crystals". Acta Crystallographica Section A: Foundations and Advances. 77 (4): 242–256. doi:10.1107/S2053273321003259. PMC 8248891.
- ↑ Fabrykiewicz, Piotr (1 July 2023). "A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions". Acta Crystallographica Section A: Foundations and Advances. 79 (4): 381–384. doi:10.1107/S2053273323003303. PMC 10317133.