---
created: 2021-12-06T01:48
updated: 2024-01-02T18:20
---

Notes on [Schubert varieties](Unsorted/Schubert.md) and [equivariant K theory](Unsorted/equivariant%20K%20theory.md)
![Pasted image 20211206015123.png](Pasted%20image%2020211206015123.png)

![Pasted image 20211206015158.png](Pasted%20image%2020211206015158.png)

![Pasted image 20211206015240.png](Pasted%20image%2020211206015240.png)

The Demazure operator:
![Pasted image 20211206015255.png](Pasted%20image%2020211206015255.png)
![Pasted image 20211206015310.png](Pasted%20image%2020211206015310.png)

![Pasted image 20211206015410.png](Pasted%20image%2020211206015410.png)
![Pasted image 20211206015440.png](Pasted%20image%2020211206015440.png)

See [Coxeter group](Coxeter%20group.md), [loop group](loop%20group), [Kac-Moody group](Kac-Moody%20group)

![Pasted image 20211206015942.png](Pasted%20image%2020211206015942.png)

![Pasted image 20211206020111.png](Pasted%20image%2020211206020111.png)

On [Kac-Moody groups](Kac-Moody%20group):
![Pasted image 20211206020154.png](Pasted%20image%2020211206020154.png)
![Pasted image 20211206020211.png](Pasted%20image%2020211206020211.png)

![Pasted image 20211206020337.png](Pasted%20image%2020211206020337.png)

![Pasted image 20211206020438.png](Pasted%20image%2020211206020438.png)

# Relation to spectra

See [equivariant homotopy theory](Unsorted/equivariant%20homotopy%20theory.md).
![Pasted image 20211206020505.png](Pasted%20image%2020211206020505.png)

![Pasted image 20211206020733.png](Pasted%20image%2020211206020733.png)
![Pasted image 20211206020742.png](Pasted%20image%2020211206020742.png)

# Positivity conjectures

Main problem in the field:
![Pasted image 20211206021355.png](Pasted%20image%2020211206021355.png)
![Pasted image 20211206021429.png](Pasted%20image%2020211206021429.png)

![Pasted image 20211206021451.png](Pasted%20image%2020211206021451.png)
![Pasted image 20211206021538.png](Pasted%20image%2020211206021538.png)
![Pasted image 20211206021559.png](Pasted%20image%2020211206021559.png)
![Pasted image 20211206022841.png](Pasted%20image%2020211206022841.png)

Using the [Bousfield-Kan spectral sequence](Bousfield-Kan%20spectral%20sequence):
![Pasted image 20211206024502.png](Pasted%20image%2020211206024502.png)

![Pasted image 20211206024630.png](Pasted%20image%2020211206024630.png)