Theori Cynrychioliad a Hynodion Cyniferydd Symplegol


Theori Cynrychioliad a Hynodion Cyniferydd Symplegol
Gwyn Bellamy

Mae rhan gyntaf yr erthygl hon yn gyflwyniad anffurfiol i theori cynrychioliadau (representation theory) y grŵp cymesur (symmetric group). Mae’r erthygl wedi ei hanelu at y mathemategydd cyffredin nad yw’n gwybod unrhyw beth am theori cynrychioliadau. Yn yr ail ran, rydym yn esbonio, yn fwy cyffredinol, sut y gellir defnyddio theori cynrychioliadau i astudio hynodion cyniferydd symplectig (symplectic quotient singularities). Yn wir, gallwn ddefnyddio theori cynrychioliadau i benderfynu pan fo’r gofodau hynod hyn yn derbyn cydraniad crepant (crepant resolution).


Cyfeiriad:

 
  	Gwyn Bellamy, ‘Theori Cynrychioliad a Hynodion Cyniferydd Symplegol’, Gwerddon, 29, Hydref 2019, 81–98.
   

Allweddeiriau

 
    Theori cynrychioliadau, grŵp cymesur, hynodion cyniferydd, cydraniad crepant, grwpiau meidraidd, geometreg algebraidd.
    

Llyfryddiaeth:

 
  	
  1. Bellamy, G. (2016), ‘Counting resolutions of symplectic quotient singularities’, Compositio Mathematica, 152 (1), 99–114.
  2. Bellamy, G., a Schedler, T. (2013), ‘A new linear quotient of C4 admitting a symplectic resolution’, Mathematische Zeitschrift, 273 (3–4), 753–69.
  3. Bellamy, G., a Schedler, T. (2016), ‘On the (non)existence of symplectic resolutions of linear quotients’, Mathematical Research Letters, 23 (6), 1537–64.
  4. Brauer, R., a Nesbitt, C. (1941), ‘On the modular characters of groups’, Annals of Mathematics (2), 42, 556–90.
  5. Brown, K. A., a Gordon, I. (2003), ‘Poisson orders, symplectic reflection algebras and representation theory’, Journal für die reine und angewandte Mathematik, 559, 193–216.
  6. Cohen, A. M. (1980), ‘Finite quaternionic reflection groups’, Journal of Algebra, 64 (2), 293–324.
  7. Curtis, C.W. (1999), Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer, cyfrol 15 o gyfres History of Mathematics (American Mathematical Society, Providence, RI; London Mathematical Society, London).
  8. Donten-Bury, M., a Wi´sniewski, J. A. (2017), ‘On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32’, Kyoto J. Math., 57 (2), 395–434.
  9. Etingof, P., a Ginzburg, V. (2002), ‘Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism’, Invent. Math., 147 (2), 243–348.
  10. Ginzburg, V., a Kaledin, D. (2004), ‘Poisson deformations of symplectic quotient singularities’, Adv. Math., 186 (1), 1–57.
  11. Gordon, I. G. (2003), ‘Baby Verma modules for rational Cherednik algebras’, Bull. London Math. Soc., 35 (3), 321–36.
  12. Hartshorne, R. (1977), ‘Algebraic Geometry’, Graduate Texts in Mathematics, 52 (New York: Springer-Verlag).
  13. Humphreys, J. E. (1972), Introduction to Lie Algebras and Representation Theory, cyfrol 9 o gyfres Graduate Texts in Mathematics (New York: Springer-Verlag).
  14. James, G. (1990), ‘The decomposition matrices of GLn(q) for n ≤ 10’, Proc. London Math. Soc. (3), 60 (2), 225–65.
  15. James, G., a Liebeck, M. (2001), Representations and characters of groups, ail argraffiad (New York: Cambridge University Press).
  16. Lehn, M., a Sorger, C. (2010), ‘A symplectic resolution for the binary tetrahedral group’, S´eminaires et Congres, 25, 427–33.
  17. Morris, A. O., a Barker, C. C. H. (1983), ‘Obituary: Dudley Ernest Littlewood’, Bull. London Math. Soc., 15 (1), 56–69.
  18. Nakajima, H. (1999), Lectures on Hilbert Schemes of Points on Surfaces, cyfrol 18 o University Lecture Series (Providence, RI: American Mathematical Society).
  19. Namikawa, Y. (2011), ‘Poisson deformations of affine symplectic varieties’, Duke Math. J., 156 (1), 51–85. London Math. Soc., 35 (3), 321–36.
  20. Hartshorne, R. (1977), ‘Algebraic Geometry’, Graduate Texts in Mathematics, 52 (New York: Springer-Verlag).
  21. Humphreys, J. E. (1972), Introduction to Lie Algebras and Representation Theory, cyfrol 9 o gyfres Graduate Texts in Mathematics (New York: Springer-Verlag).
  22. James, G. (1990), ‘The decomposition matrices of GLn(q) for n ≤ 10’, Proc. London Math. Soc. (3), 60 (2), 225–65.
  23. James, G., a Liebeck, M. (2001), Representations and characters of groups, ail argraffiad (New York: Cambridge University Press).
  24. Lehn, M., a Sorger, C. (2010), ‘A symplectic resolution for the binary tetrahedral group’, S´eminaires et Congres, 25, 427–33.
  25. Morris, A. O., a Barker, C. C. H. (1983), ‘Obituary: Dudley Ernest Littlewood’, Bull. London Math. Soc., 15 (1), 56–69.
  26. Nakajima, H. (1999), Lectures on Hilbert Schemes of Points on Surfaces, cyfrol 18 o University Lecture Series (Providence, RI: American Mathematical Society).
  27. Namikawa, Y. (2011), ‘Poisson deformations of affine symplectic varieties’, Duke Math. J., 156 (1), 51–85.
  28. Reid, M. (1983), ‘Minimal models of canonical 3-folds’, yn Algebraic varieties and analytic varieties (Tokyo, 1981), cyfrol 1 o Advanced Studies in Pure Mathematics (Amsterdam: North-Holland Publishing Co.), tt. 131–180.
  29. Robinson, G. de B. (1947), ‘On a conjecture by Nakayama’, Proceedings and transactions of the Royal Society of Canada. Sect. III. (3), 41, 20–25.
  30. Serre, J. P., (1977), Linear representations of finite groups (Springer-Verlag, New York-Heidelberg). Cyfieithiwyd o’r ail argraffiad yn y Ffrangeg gan Scott, Leonard L, Graduate Texts in Mathematics, Vol. 42.
  31. Verbitsky, M. (2000), ‘Holomorphic symplectic geometry and orbifold singularities’, Asian Journal of Mathematics, 4(3), 553–63.
  32. Williamson, G. (2017), ‘Schubert calculus and torsion explosion’, Journal of the American Mathematical Society, 30(4), 1023–46. Gydag atodiad gan Alex Kontorovich a Peter J. McNamara.