on a property of minuscules in weight lattice

Question

Let $\Phi$ be a root system in inner product space $E=\mathbb{R}^l$ (inner product $(\cdot,\cdot)$ and $\Delta=\{\alpha,_1,\cdots,\alpha_l\}$ be a fundamental root system.

Consider the root lattice $\Lambda_r=\mathbb{Z}\alpha_1 + \cdots + \mathbb{Z}\alpha_l$ in $E$ and weight lattice $\Lambda$ $$\Lambda =\{\lambda\in E : \frac{2(\lambda,\alpha)}{(\alpha,\alpha)} \in\mathbb{Z} \forall \alpha\in \Delta\}. $$

Let $\Lambda^+ = \{ \lambda\in\Lambda : \frac{2(\lambda,\alpha)}{(\alpha,\alpha)} \geq 0 \forall \alpha\in\Delta\}$.

The miniscules are specific elements of $\Lambda^+$; they are minimal elements of $\Lambda^+$ w.r.t ordering defined as follows: $\lambda_1\leq \lambda_2$ if $\lambda_2-\lambda_1=\sum_{i=1}^l m_i\alpha_i$ with $m_i\geq 0$.

I wanted to see the proof of following fact (seen here this, page 2, top line)

For different minuscules $\lambda,\nu$, the cosets $\lambda+\Lambda_r$ and $\nu+\Lambda_r$ are distinct.

Is this easy to prove? How should we proceed?

I tried a simple case: suppose $\lambda,\nu$ are minuscules with $\lambda=\nu+\alpha-\beta$ where $\alpha,\beta$ are distinct fundamental (i.e. simple) roots. I couldn't get any direction to get contradiction from this.

The actual general problem is that if $\lambda+\Lambda_r=\nu+\Lambda_r$, then $\lambda-\nu$ is integral combination of fundamental roots, so y $\lambda=\nu+m_1\alpha_1 + \cdots + m_l\alpha_l$, so some coefficients could be positive and some negative. The simplest non-trivial case is then exactly one $m_i$ is $+1$, exactly one $m_j$ is $-1$ and rest are zero; I don't get any idea to proceed for contradiction.

Answer 1

To comment on the question here (in community-wiki mode), I should point out first that $\S13$ of my now-ancient book was meant to develop some properties of weights just in the framework of abstract root systems without having developed the representation theory where the ideas first arose. In this experimental spirit, my exercises were a bit speculative (and in some cases not worked out by me in detail until much later, since I knew the conclusions were correct, e.g., Exercise 13.10).

That being said, my Exercise 13.13 is intended to be done using just the tools of this chapter. However, the exercise itself comes from one in Bourbaki, VI, $\S2$, which I've now added (reluctantly) to the informal notes linked in the question. The Bourbaki exercise relies on their development in Chapter V of ideas about reflection groups, applied to affine Weyl groups. This gets complicated, since it uses the dual root system in an essential way. I didn't go into affine Weyl groups in my 1972 book but did include an account in Chapter 4 of my 1990 book on reflection groups. An advantage of that setting is to visualize things better: the affine Weyl group leads to a euclidean simplex (fundamental alcove) whose nonzero vertices are of the form $\varpi/c$ for the fundamental weights $\varpi$ and corresponding coefficients $c$ in the highest root of the (dual) root system. Only for $c=1$ do you get minuscule weights (none in some types of irreducible root system).

Anyway, one is led to the list in my Exercise 13.13 (or in Bourbaki's Chapter VIII). A subtle point is the dualization, which assigns (in Bourbaki's numbering which I adopted) $\lambda_\ell$ to type $B_\ell$ and $\lambda_1$ to type $C_\ell$. Bourbaki originally reversed these, but that seems to be corrected in the English translation. (Note too that they used E. Cartan's symbol $\varpi$, a version of the handwritten $\pi$ for poids, whereas I used $\lambda$. Their notation is more commonly used but is easy to confuse with $\omega$.)

Finally, a comment on the spelling of minuscule. This is the original version, emphasizing MINUS (I guess in contrast to majuscule), but often rendered as miniscule with emphasis on MINI. The question here mixes both spellings, which is undesirable.

Answer 2

Possibly this argument works; I am also new in Lie groups and Lie algebras, so I am happy for experts comments.

Consider $\lambda$ a minuscule i.e. a minimal dominant weight (according to Humphreys).

Let $\{\lambda_1,\cdots,\lambda_l\}$ be dual basis of $\{\check{\alpha_1},\cdots,\check{\alpha_l}\}$ w.r.t. $(\cdot,\cdot)$.

1. $\lambda + m_1\alpha_1 + \cdots m_l\alpha_l $, for $m_i\geq 0$, is not minimal dominant weight unless all $m_i=0$.

Here all $m_i\geq 0$. If some $m_i>0$ then $\lambda'=\lambda + m_1\alpha_1 + \cdots m_l\alpha_l $ is bigger than $\lambda$ (i.e. by def.,$\lambda'-\lambda$ is sum of fundamental roots). Then minimality of both $\lambda$ and $\lambda'$ (as dominant weights) implies that they should be equal so $m_1\alpha_1 + \cdots m_l\alpha_l =0$; but $\alpha_i$'s are independent and one $m_i$ is non-zero, so contradiction.

2. Every dominant weight can be written as $\sum q_i\alpha_i$ where $q_i$ are positive rationals.

If $\lambda$ is dominant weight then this means $\lambda=c_1\lambda_1 + \cdots + c_l\lambda_l$ where $c_i\in\mathbb{Z}^+$. Further, each $\lambda_i$ can be written as $a_{i1}\alpha_1 + \cdot + a_{il}\alpha_l$ where $a_{ij}$ are positive rationals (see Exercise 13.8 in Humphreys Lie algebra). This proves the assertion (2).

3 (Main fact:) Every minimal dominant weight $\lambda$ can be written as $\sum q_i\alpha_i$ where $q_i$'s are positive rationals, less than $1$.

Write a dominant weight $\lambda$ as $\sum q_i'\alpha_i$ with $q_i'$ positive rationals. If some $q_j'>1$ then we can subtract $\alpha_j$ from $\lambda$ to get a smaller dominant weight.

4. If $\lambda,\nu$ are minimal dominant weights in same coset of $\Lambda_r$, then $\lambda-\mu$ is integral combination of $\alpha_i$'s; so $$\lambda=\nu + c_1\alpha_1 + \cdots + c_l\alpha_l, (c_i\in\mathbb{Z}).$$

If all $c_i's$ are $\geq 0$, apply $1$; if all $c_i\leq 0$, then take them to left side and apply 1.

If some $c_i>0$ then in RHS, coefficient of $\alpha_i$ will be rational $>1$, but in LHS, it is less than $1$, contradiction to (3).