Some "homework" exercises.

Let (W,S) be a Coxeter system, R = {roots}, R+ = {positive roots} = {a \in R | a>0} (this >0 is just convenient notation); Pi = {a_s | s \in S} = {simple roots}.

Set R(w) = { a > 0 | w(a) < 0}. Set l(w) = |R(w)| .

(a) If a_s \not\in R(w) [i.e. w(a_s) > 0] , verify l(ws) = l(w) + 1. (One symbol is an ell, the other a one).

In this case, we sometimes just write ws > w.

Describe R(ws) in terms of a_s, s, and R(w).

(b) If w^{-1}(a_s) > 0 [and so necessarily w^{-1}(a_s) \not\in R(w); if we had w^{-1}(a_s) < 0, then we get -w^{-1}(a_s) = w^{-1}(-a_s) \in R(w)], verify l(sw) = l(w) + 1. Describe R(sw) in terms of a_s, w, and R(w).

(a') If a_s \in R(w) , verify l(ws) = l(w) - 1. Describe R(ws) in terms of a_s, s, and R(w).

(b') If w^{-1}(a_s) < 0 , verify l(sw) = l(w) - 1. Describe R(sw) in terms of a_s, w, and R(w).

(c) If w = s_{i_k} ... s_{i_2} s_{i_1} with s_{i_j} \in S and with k minimal, verify k = l(w). Such an expression is called a reduced word expression for w. It is NOT unique, but k is independent of reduced word. So many people use this notion (length of a reduced word) as the definition of length. Write R(w) in terms of the a_j = a_{s_j} and the s_{i_j}.

(d) Check: if l(wv) = l(w) + l(v) , then R(wv) = R(v) union v^{-1} R(w). (Conversely, v^{-1} (a) > 0 for all a \in R(w) implies that l(wv) = l(w) + l(v). )

3. Read in Humphreys 1.10 Parabolic subgroups and minimal coset representatives

4. Read in Humphreys 4.1, 4.2 Affine reflections, affine Weyl groups

5. Read in Humphreys 5.1 - 5.11 for general Coxeter groups, length, roots, parabolic subgroups, Bruhat order

6. KL polys for the dihedral group.

D_m = <s,t | (st)^m = 1 > = <s,t | stst .... = tsts ... (where there are m terms on each side of the =) >

(geometrically, picutre two lines in the plane making an angle pi/m with each other (acute; the obtuse angle is pi- pi/m), s is reflection over one line, t is reflection over the other, so that their composition st is a rotation by 2pi/m (hence has multiplicative order m).

As a Coxeter group, S={s,t} = {s_1, s_2}, W = D_m.

(a) Construct the corresp Iwahori-Hecke algebra over Z[q].

(generators T_i, with T_w = T_i T_j ... T_k if w = s_i s_j ... s_k is reduced) and relations

T_i T_w = { T_{s_i w} if s_i w > w,

{ q T_{s_i w} + (q-1) T_w if s_i w < w.

(b) Construct its KL basis

(c) Hence, compute the assoc KL polys

(d) When m is even, you can pick unequal parameters (q_s, q_t). What happens then?)

7. KL polys for the symmetric group. (just for n=3)

S_3 = <s,t | sts = tst >

As a Coxeter group, S={s,t} = {s_1, s_2}, W = S_3.

(a) Construct the corresp Iwahori-Hecke algebra over Z[q].

(generators T_i, with T_w = T_i T_j ... T_k if w = s_i s_j ... s_k is reduced) and relations

T_i T_w = { T_{s_i w} if s_i w > w,

{ q T_{s_i w} + (q-1) T_w if s_i w < w.

(b) Construct its KL basis

(c) Hence, compute the assoc KL polys

(d) If you are feeling adventurous, do this for S_4 too.

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1. Read in Humphreys 7.4- 7.9.

Do ex 1 in 7.3, ex pg 150 in 7.4, ex 1 , ex 3 in 7.5 , ex pg 156 in 7.7, ex pg 157 in 7.8, ex pg 160 of 7.10

2. Read in Humphreys 7.13, maybe 7.14, 7.15

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1. Pick a k and n of choice (n > 2, k> 1), and compute the global basis and the action of the crystal operators on the crystal basis explicitly , for U_q(sl_k) acting on the n-th tensor product of V = C^k.

Note, we did most of this in class for k=3, n=2.

2. (a) Pick a k (k= 3 is best to get a feel for this), and compute the global basis and the action of the crystal operators on the crystal basis explicitly , for U_q(\hat sl_k) acting on the basic representation (that sits in the Fock space), for a few small partitions \mu. Recall, \mu will be a k-regular partition.

(b) The change of basis coefficients, d_{\lambda,\mu}(q) are affine parabolic KL polys. In case you used this to compute the G(\mu), now just use the LLT algorithm (or some ad hoc variant that computes the G(\mu) using their bar invariance and triangularity) to compute them and compare.

If that was already how you computed them, now compute the appropriate affine parabolic KL polys and compare.

(c) When k is large and \mu has few boxes, the G(\mu) should be nice. For the same \mu as above, computer some G(\mu) and compare.