On Groebner bases and immersions of Grassmann manifolds $G_{2,n}$

Abstract

Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal. A Groebner basis for this ideal is obtained. Using this basis, some new immersion results for Grassmannians G 2,n are established.

Groebner Bases and Nonembeddings of Some Flag Manifolds

“…Note that G corresponds to the Groebner basis for the ideal determining the Z 2 -cohomology of Grassmann manifold G 2,n+1 obtained in [10], while the set G defined in [10, page 115] is the Groebner basis for the corresponding ideal for the Grassmannian G 2,n . Obviously, g 0 = f n+2 , and also, it is not hard to verify the relations y 1 g 0 + g 1 = f n+3 and y 2 g m + y 1 g m+1 = g m+2 , m = 0, n [10, page 115].…”

“…In [10] it is shown that the set G is the reduced Groebner basis for the ideal (G) Z 2 [y 1 , y 2 ] (which determines the Z 2 cohomology of the Grassmannian G 2,n+1 ) with respect to the grlex ordering in Z 2 [y 1 , y 2 ] (y a 1 y b 2 grlex y c 1 y d 2 if and only if a + b < c + d or else a + b = c + d and a c). We wish to define a term ordering in Z 2 [x, y 1 , y 2 ] which restricts to this grlex ordering in Z 2 [y 1 , y 2 ].…”

“…Also, no term of g m ∈ G is divisible by the leading term of some g l ∈ G \ {g m } since G is reduced [10,Theorem 2.7]. Since gcd(LT(z n+1 ), t) = 1 for all t ∈ T (g m ) and all g m ∈ G, no term of g m ∈ G is divisible by LT(z n+1 ).…”

Multiplication in the cohomology of Grassmannians via Gröbner bases

“…In various applications it is important to determine if a certain cohomology class, given in terms of Stiefel-Whitney classes, is zero or not -for example, in determining the span of Grassmannians, in discussing immersions and embeddings in Euclidean spaces, in the determination of cup-length (which is related to the Lusternik-Schnirelmann category), in some geometrical problems which may be reduced to the question of the existence of a non-zero section of a bundle over a Grassmann manifold, etc. It is evident from [6,9,12], that having a Gröbner basis for the ideal that determines H * (G k,n (R); Z 2 ), can be very helpful for answering these kind of questions. In this paper, using the corresponding result for complex Grassmannians, we obtain Gröbner bases for these ideals and give an application to the immersion problem for the manifolds G 5,n (R).…”

“…In [4] and [5] recurrence formulas of similar flavour were obtained for inverse Kostka matrix. In Section 6, using the results for the complex Grassmannians, we construct Gröbner bases for the ideals that, by Borel's description, determine the Z 2 -cohomology of real Grassmannians, thus completing the research done in [9] and [12]. Finally, in Section 7 we apply the results for real Grassmann manifolds (obtained in Section 6) and prove an immersion theorem which establishes new immersions for an infinite family of these manifolds (in particular, G 5,n (R) when n is a power of two).…”