By Bibhutibhushan Datta
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Extra resources for Ancient Hindu Geometry: The Science Of The Sulba
It allows us to express i1 : : : ir ] (i+1)1 : : : (i+1)r ] as a sum of the products i1 : : : ^is : : : ir (i+1)j ] (i+1)1 : : : ^(i+1)j : : : (i+1)r is ]: CHAPTER 2. THE FIRST FUNDAMENTAL THEOREM 24 Substituting this in the product of the bracket functions corresponding to the rows of , we express as a sum of the 0 such that the mark of each 0 is greater than the mark of (with respect to the lexicographic order). Continuing as a sum of standard tableau functions. in this way we will be able to write This shows that the standard tableau functions span the space Tab r m (w ).
However, if G is irreducible, the orbit O(x) is a locally closed subset of X , and hence is a quasi-projective algebraic variety. It follows from the Chevalley Theorem (see , p. 94), that the image of a regular map is a disjoint finite union of locally closed subsets. , a quasi-projective variety. Of course, the image of an affine variety is not always affine. 3. Let H be a closed subgroup of an algebraic group G. Consider the space V of (G) spanned by the G-translates of generators of the ideal I defining H .
Since F (v1 : : : vn ) 2 AG (because F is G-invariant), we are done. Now we are ready to finish the proof of Nagata’s Theorem. To begin, by noetherian induction, we may assume that for any nontrivial G-invariant ideal I the algebra (A=I )G is finitely generated. , A0 = k) and that the action of G preserves the grading. For example, A could be a polynomial algebra on which G acts linearly. The subalgebra A G inherits the P CHAPTER 3. REDUCTIVE ALGEBRAIC GROUPS 44 2 grading. Suppose AG is an integral domain.
Ancient Hindu Geometry: The Science Of The Sulba by Bibhutibhushan Datta