By Thomas Lam, Luc Lapointe, Jennifer Morse, Mark Shimozono
Quantity 208, quantity 977 (second of 6 numbers).
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Extra info for Affine insertion and Pieri rules for the affine Grassmannian
19 v(b1 ) ≤ v(j) − n < v(j) < v(i) < v(j) + n . 2. Therefore, (ii) holds. 30) follows. 15. In Case C, u(i) > u(b1 ). 36) Proof. 6(iv). 19, w(i) > w(b1 ). 14, we have either (i) ¯b1 = ¯j or (ii) j = b1 +kn, with both implying u(j) > u(b1 ). Since C is a strong cover, u(i) > u(j), so that u(i) > u(b1 ) as desired. 16. 38) y −1 (q) ≤ l < y −1 (p). Moreover q is A-bad and p is A -bad. Proof. 29), and that p is A -bad. 17. 40) y −1 (q) ≤ l. 30). 17. 13. 2 applied to the strong cover y y ti,b1 we have either (a) b1 −i < n or (b) y(b1 ) − y(i) < n.
Proof. We have m(C ) = x(i) = cA u(i) = cA (m(C)), which proves (i). 15. (iv) was proved above. 8. In Case A, S is a strong strip. 3. PROOFS FOR THE LOCAL RULE 31 Proof. 6) where the top row gives S and the bottom − row gives S . By induction S1 is a strong strip, so we need only check that m(C ) < m(C ), that is, v(a1 ) < v(j). Since S is a strong strip w(i− ) < w(j). 6(i) asserts that either (a) v(a1 ) < w(i− ) or (b) v(a1 ) = cA (w(i− )). Suppose (a) holds. 4), v(j) = cA (w(j))) ≥ w(j) − 1 ≥ w(i− ) > v(a1 ), as desired.
However the Pieri rules obtained from the two diﬀerent markings agree. 5. Geometric interpretation of strong Schur functions In this section we list some conjectural properties of strong Schur functions, assuming l = 0 for simplicity. 18. Let u, v ∈ S˜n be two aﬃne permutations. We have the following successively stronger properties. (1) We have Strongu/v (x) ∈ Λ. (2) We have Strongu/v (x) ∈ Λ(n) . 3). The corresponding properties of weak Schur functions are known. 3) was proven combinatorially in  (see also ) while positivity was shown in  using geometric work of Peterson .
Affine insertion and Pieri rules for the affine Grassmannian by Thomas Lam, Luc Lapointe, Jennifer Morse, Mark Shimozono
Categories: Nonfiction 13