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GLn- DECOMPOSITION OF THE SCHUR COMPLEX Sr2 φ)

Choi, Eun J.;Kim, Young H.;Ko, Hyoung J.;Won, Seoung J.

  • Published : 2003.02.01

Abstract

In this paper we construct a natural filtration associated to the plethysm $S_{r}(\wedge^2 \varphi)$ over arbitrary commutative ring R. Let $\phi$ : G longrightarrow F be a morphism of finite free R-modules. We construct the natural filtration of $S_{r}(\wedge^2 \varphi)$ as a $GL(F){\times}GL(G)$- complex such that its associated graded complex is ${\Sigma}_{{\lambda}{\in}{\Omega}_{\gamma}}=L_{2{\lambda}{\varphi}$, where ${{\Omega}_{\gamma}}^{-}$ is a set of partitions such that $│\wedge│\;=;{\gamma}\;and\;2{\wedge}$ is a partition of which i-th term is $2{\wedge}_{i}$. Specializing our result, we obtain the filtrations of $S_{r}(\wedge^2 F)\;and\;D_{r}(D_2G).㰌搀؀㘲㌮㠻Ⰰ䉵楬摩湧Ⱐ捯湳瑲畣瑩潮⁡湤⁣楶楬⁥湧楮敥物湧

Keywords

natural filtration;plethysm;$S_{r}(\wedge^2 \varphi)$;$S_{r}(\wedge^2 F)$;$D_{r}(D_2G)$

References

  1. Gesammelte Abhandlungen 1 Klassen von Matrizer, die sich einen gegebenen Matrix xuorden lassen (1901) : in I. Schur I. Schur https://doi.org/10.1073/pnas.86.8.2521
  2. The Classical Groups H. Weyl
  3. Adv. in Math. v.44 Schur functors and Schur complexes K. Akin;D. A. Buchsbaum;J. Weyman https://doi.org/10.1016/0001-8708(82)90039-1
  4. Symmetric functions and Hall polynomials (2nd ed.) I. G. Macdonald
  5. Brandeis Lecture Notes v.3 Generic free resolutions and Schur complexes D. A. Buchsbaum
  6. Adv. in Math. v.89 On some plethysms G. Boffi https://doi.org/10.1016/0001-8708(91)90075-I
  7. Trans. Amer. Math. Soc. v.324 The decomposition of Schur complexes H. J. Ko https://doi.org/10.2307/2001506
  8. Adv. in Math. v.72 Characteristic-free representation theory of the general linear group K. Akin;D. A. Buchabaum https://doi.org/10.1016/0001-8708(88)90027-8
  9. Proc. Nat. Acad. Sci. v.86 Standard basis in supersymplectic algebras G. C. Rota;J. A. Stein https://doi.org/10.1073/pnas.86.8.2521
  10. Adv. in Math. v.94 Resolutions of determinantal ideals : n- minors of (n + 2 )- square matrices M. Hashimoto;K. Kurano https://doi.org/10.1016/0001-8708(92)90032-G
  11. Adv. in Math. v.35 Young diagrams and ideals of Pfaffians S. Abeasis;A. Del Fra https://doi.org/10.1016/0001-8708(80)90046-8
  12. Philos. Trans. Roy. Soc. London Ser. v.A239 Invariant theory, tensors, and group characters D. E. Littlewood https://doi.org/10.1016/0021-8693(89)90139-7
  13. J. Algebra v.124 On relations on minors of generic symmetric matrices K. Kurano https://doi.org/10.1016/0021-8693(89)90139-7
  14. The theory of group characters (2nd ed.) D. E. Littlewood https://doi.org/10.1098/rsta.1944.0001
  15. Adv. in Math. v.58 Characteristic-free representation theory of the general linear group K. Akin;D. A. Buchabaum https://doi.org/10.1016/0001-8708(85)90115-X