Maximal indexes of flag varieties for spin groups

Abstract

We establish the sharp upper bounds on the indexes for most of the twisted flag varieties under the spin groups ${\operatorname {\mathrm {Spin}}(n)}$ .

Indexes of generic Grassmannians for spin groups

“…Note that the mentioned result of [2] and the determination of 𝑡 in Example 1.2 (up to 'plus 1') are the special cases of this inequality.…”

“…(2) finding a strong lower bound on i(𝑚) which is its exact value for most 𝑑, 𝑚 and which is always within 1 from the exact value; (3) determination of 𝑑 and 𝑚 such that i(𝑚) coincides with the lower bound obtained in (2).…”

“…where g 𝑘 is a polynomial with integer coefficients in 𝑧, 𝑦 Note that the ring 𝑅 𝑊 is generated by the elementary symmetric polynomials in 𝑥 1 , … , 𝑥 𝑚 and the elementary symmetric polynomials in 𝑦 2 1 , … , 𝑦 2 𝑙 . Also note that all 𝑓 𝑘 and z are in the 𝑊invariant subring 𝑅 0 [𝑧] ⊂ 𝑅 [𝑧], where 𝑅 0 ∶= ℤ[𝑦 1 , … , 𝑦 𝑙 ].…”

“…Note that the ring 𝑅 𝑊 is generated by the elementary symmetric polynomials in 𝑥 1 , … , 𝑥 𝑚 , the elementary symmetric polynomials in 𝑦 2 1 , … , 𝑦 2 𝑙 , and the product 𝑦 1 ⋯ 𝑦 𝑙 . Note that the last generator (and only it) did not show up in § 4.…”

“…The images in CH(𝐺∕𝑃) of the elementary symmetric polynomials in 𝑥 1 , … , 𝑥 𝑚 are the Chern classes of  . The images of the elementary symmetric polynomials in 𝑦 2 1 , … , 𝑦 2 𝑙 are the Chern classes of the quotient  ⊥ ∕ . The isomorphism 𝑉∕ ⊥ =  ∨ , where  ∨ is the dual vector bundle, shows that the Chern classes of  ⊥ are polynomials in the Chern classes of  .…”