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Environment to perform calculations of equivariant vector bundles on homogeneous varieties

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License: GPL3
ubuntu2204
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        Initialize the Grassmannian Gr(k,N).

        INPUT:
        - ``k`` -- Integer ;
        - ``N`` -- Integer ;

        OUTPUT: None.
        z%The input for `N` must be an integer.z3The integer `N` must be equal to or greater than 2.z%The input for `k` must be an integer.z(The integer `k` need to be in the range �.r)�Irreducible_Cartan_Group�A)�
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�|�d�$�$�-�-�q�2�2�2�2�2�c�R�|���|���fS�zDReturns all attributes which are necessary to initialize the object.�r	r
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�/Returns a one-line string as short description.�ShortzGr(�;�)r-zGrassmannian variety of z#-dimensional linear subspaces in a z"-dimensional ambient vector space.�0The input for ``Output_Style`` is inappropriate.)rr	r
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        Returns Fonarev's (conjecturally full) minimal exceptional collection on ``self``.

        OUTPUT:
        - (Conjecturally full) minimal exceptional collection

        REFERENCE:
        - [Fon2012] Fonarev, A.: On minimal Lefschetz decompositions for Grassmannians
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        Returns Kapranov's full exceptional collection on ``self``.

        OUTPUT:
        - Full exceptional collection cU^alpha with n-k => alpha_1 => alpha_2 => ... => alpha_k => 0 (lexicographically orderd)

        REFERENCE:
        - [Kap1988] Kapranov, M. M.: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math.92(1988), no.3, 479–508.
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        Initialize the projective space of a given dimension.

        INPUT:
        - ``Dimension`` -- Integer ;

        OUTPUT: None.
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        Returns Beilinson's full exceptional collection on ``self``.

        OUTPUT:
        - Lefschetz collection with starting block ( O_X ) and support partition (self.Dimension()+1)*[ 1 ].

        REFERENCE:
        - [Bei1978] Beilinson, A. A.: Coherent sheaves on Pn and problems in linear algebra. Funktsional. Anal. i Prilozhen.12(1978), no.3, 68–69.
        r=)�calOrDrirr9)r"r>r?s   r&�Beilinson_Collectionz%Projective_Space.Beilinson_Collection|sQ�� �9�9�;�;����$�.�.�"2�"2�=�"@�]�CU�!U�V�V���(�(��Zi�(�k�k�kr'r_r`)
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        Initialize the dual projective space of a given dimension.

        INPUT:
        - ``Dimension`` -- Integer ;

        OUTPUT: None.
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