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# 数学代写|组合数学代写Combinatorial Mathematics代考|MATH4306 Study of ffnite subsets on other spaces

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## 数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on projective spaces (compact symmetric spaces of rank 1)

The study of finite subsets on the sphere is generalized to the study of finite subsets on the real projective space. The real projective space is obtained by identifying two antipodal points of the sphere. Therefore, we can regard the study of finite subsets on the real projective space as the study of antipodal subsets on the sphere. In this sense, some of the results mentioned in the previous sections can be regarded as the results on the real projective space. The projective spaces over the real and complex spaces are identified as the space of 1-dimensional subspaces of the vector space over each of these fields. In this sense, the study of equiangular lines has been made. See Delsarte, Goethals, and Seidel (1975) [162] and Koornwinder (1976) [286]. Besides real and complex fields, there are projective spaces over the quaternion (skew) field and the Cayley octonion. (Only the projective plane exists over the Cayley octonion.) These spaces are, together with the sphere, called compact symmetric spaces of rank 1 . It is shown by Cartan [124] (work in the 1920s) that these are the only connected compact symmetric spaces of rank 1. Also, H.-C. Wang (1952) [503] showed that the compact symmetric spaces of rank 1 are characterized as compact 2-point homogeneous symmetric spaces. Note that this last property essentially corresponds to the distance-transitive property, and therefore in the case of graphs, to the property of distance-regular graphs or Ppolynomial association schemes.

## 数学代写|组合数学代写Combinatorial Mathematics代考|Finite subsets on compact symmetric spaces of general ranks

The compact symmetric spaces of a general rank were classified by E. Cartin around 1920 Helgason (1962) [213], Wolf [519], mathematical dictionaries, etc.

Now, we will see that finite subsets (codes and designs) on these spaces are very much studied. Typical compact symmetric spaces of rank greater than 1 are real Grassmann spaces. Finite subsets there have been studied extensively by Shor, Sloane, Calderbank, Hardin, Rains, and others. This work came from the study of error correcting codes in quantum computers, indeed a very interesting topic. The reader is referred to $[427,118]$. For a simple explanation in Japanese, please see [32, Chapter 15, Section 15.6].

The study of finite subsets of real Grassmann spaces from the viewpoint of algebraic combinatorics was started by Bachoc, Coulangeon, and Nebe (2002) [10], and then further developed in Bachoc, Bannai, and Coulangeon (2004) [9]. There, the concepts of $t$-designs and tight $t$-designs were defined, and a similar theory to the case of those on the spheres has been developed, although the classification of tight $t$-designs is not yet within reach. In the case of compact rank 1 symmetric spaces, 1-variable Jacobi polynomials appear as spherical functions. In the case of real Gassmann space, multivariable orthogonal polynomials appear. In fact, in the general compact symmetric rank $r$, certain generalized Jacobi polynomials of $r$ variables appear (the reader is referred to Vretare (1984) [498] or further more general work by Koornwinder (cf. [285])).

Roughly speaking, there are two kinds of compact symmetric spaces. One class consists of the homogeneous spaces $G / H$ of simple Lie groups $G$ by subgroups $H$, and another class consists of the compact simple Lie groups $G$ themselves. (The second class is also interpreted as $(G \times G) / G$ by the diagonal subgroup isomorphic to $G$.) The real Grassmann space is one in the first class, and the work on the real Grassmann case is generalized for other symmetric spaces in the first class. (For the complex Grassmann case, see Y. Miura [350] [the Master’s thesis of Kyushu University (2004)] and Roy (2010) [406]. There was an unpublished generalization by Takanori Yasuda on this topic.)

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