### D.15.3 chern_lib

Library:
  chern.lib
Purpose:
  Symbolic Computations with Chern classes

Author:
  Oleksandr Iena, oleksandr.iena@uni.lu, yena@mathematik.uni-kl.de

References:
  [1] Iena, Oleksandr, On symbolic computations with Chern classes: remarks on the library chern.lib for Singular,
  http://hdl.handle.net/10993/22395, 2015
  [2] Lascoux, Alain, Classes de Chern d'un produit tensoriel. C. R. Acad. Sci., Paris, Ser. A 286, 385-387 (1978).

Procedures:

> D.15.3.1 symm |  | symmetric functions in the entries of l
> D.15.3.2 symNsym |  | symmetric and non-symmetric parts of a polynomial f
> D.15.3.3 CompleteHomog |  | complete homogeneous symmetric functions
> D.15.3.4 segre |  | Segre classes in terms of Chern classes
> D.15.3.5 chern |  | Chern classes in terms of Segre classes
> D.15.3.6 chNum |  | the non-zero Chern numbers in degree N in the entries of c
> D.15.3.7 chNumbers |  | the Chern numbers in degree N in the entries of c
> D.15.3.8 sum_of_powers |  | the sum of k-th powers of the entries of l
> D.15.3.9 powSumSym |  | the sums of powers [up to degree N] in terms of the elementary symmetric polynomials (entries of l)
> D.15.3.10 chAll |  | Chern character in terms of the Chern classes
> D.15.3.11 chAllInv |  | Chern classes in terms of the Chern character
> D.15.3.12 chHE |  | the highest term of the Chern character
> D.15.3.13 ChernRootsSum |  | the Chern roots of a direct sum
> D.15.3.14 chSum |  | the Chern classes of a direct sum
> D.15.3.15 ChernRootsDual |  | the Chern roots of the dual vector bundle
> D.15.3.16 chDual |  | the Chern classes of the dual vector bundle
> D.15.3.17 ChernRootsProd |  | the Chern roots of a tensor product of vector bundles
> D.15.3.18 chProd |  | Chern classes of a tensor product of vector bundles
> D.15.3.19 chProdE |  | Chern classes of a tensor product of vector bundles
> D.15.3.20 chProdL |  | Chern classes of a tensor product of vector bundles
> D.15.3.21 chProdLP |  | total Chern class of a tensor product of vector bundles
> D.15.3.22 ChernRootsHom |  | the Chern roots of a Hom vector bundle
> D.15.3.23 chHom |  | Chern classes of the Hom-vector bundle
> D.15.3.24 ChernRootsSymm |  | the Chern roots of the n-th symmetric power of a vector bundle with Chern roots from l
> D.15.3.25 ChernRootsWedge |  | the Chern roots of the n-th exterior power of a vector bundle with Chern roots from l
> D.15.3.26 chSymm |  | the rank and the Chern classes of the k-th symmetric power of a vector bundle of rank r with Chern classes c
> D.15.3.27 chSymm2L |  | the rank and the Chern classes of the second symmetric power of a vector bundle of rank r with Chern classes c
> D.15.3.28 chSymm2LP |  | the total Chern class of the second symmetric power of a vector bundle of rank r with Chern classes c
> D.15.3.29 chWedge |  | the rank and the Chern classes of the k-th exterior power of a vector bundle of rank r with Chern classes c
> D.15.3.30 chWedge2L |  | the rank and the Chern classes of the second exterior power of a vector bundle of rank r with Chern classes c
> D.15.3.31 chWedge2LP |  | the total Chern class of the second exterior power of a vector bundle of rank r with Chern classes c
> D.15.3.32 todd |  | the Todd class
> D.15.3.33 toddE |  | the highest term of the Todd class
> D.15.3.34 Bern |  | the second Bernoulli numbers
> D.15.3.35 tdCf |  | the coefficients of the Todd class of a line bundle
> D.15.3.36 tdTerms |  | the terms of the Todd class of a line bundle coresponding to the Chern root t
> D.15.3.37 tdFactor |  | the Todd class of a line bundle coresponding to the Chern root t
> D.15.3.38 cProj |  | the total Chern class of (the tangent bundle on) the projective space P_n
> D.15.3.39 chProj |  | the Chern character of (the tangent bundle on) the projective space P_n
> D.15.3.40 tdProj |  | the Todd class of (the tangent bundle on) the projective space P_n
> D.15.3.41 eulerChProj |  | Euler characteristic of a vector bundle on the projective space P_n via Hirzebruch-Riemann-Roch theorem
> D.15.3.42 chNumbersProj |  | the Chern numbers of the projective space P_n
> D.15.3.43 classpoly |  | polynomial in t with coefficients from l (without constant term)
> D.15.3.44 chernPoly |  | Chern polynomial (constant term 1)
> D.15.3.45 chernCharPoly |  | polynomial in t corresponding to the Chern character (constant term r)
> D.15.3.46 toddPoly |  | polynomial in t corresponding to the Todd class (constant term 1)
> D.15.3.47 rHRR |  | the main ingredient of the right-hand side of the Hirzebruch-Riemann-Roch formula
> D.15.3.48 SchurS |  | the Schur polynomial corresponding to partition I in terms of the Segre classes S
> D.15.3.49 SchurCh |  | the Schur polynomial corresponding to partition I in terms of the Chern classes C
> D.15.3.50 part |  | partitions of integers not exceeding n into m non-negative summands
> D.15.3.51 dualPart |  | partition dual to I
> D.15.3.52 PartC |  | the complement of a partition with respect to m
> D.15.3.53 partOver |  | partitions over a given partition J with summands not exceeding n
> D.15.3.54 partUnder |  | partitions under a given partition J

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