Last week Timothy discussed the Kazhdan-Lusztig polynomials for Harish-Chandra modules. Suppose p = (x,lambda,nu) \ne p' = (x',lambda',nu') are parameters for distinct Harish-Chandra modules, say with regular integral infinitesimal character (just to isolate the most interesting behavior). Then there's a KL polynomial P = P_{p',p} which is a polynomial (say in q) with nonnegative integer coefficients, of degree at most (dimension(x) - dimension(x') - 1)/2. (The "-1" is present because of the assumption that the parameters are distinct.) Kazhdan and Lusztig singled out the "top degree term" mu(p',p) = coeff of q^{(dimension(x) - dimension(x') - 1)/2}. They used these integers to define a "W-graph" (a very interesting representation of the Hecke algebra, and even of W, with basis the irreducible HC modules). "Most" of the mu's are zero. If you know all the mu's, then calculating the rest of the KL polynomials is pretty easy/fast; and in any case the mu's tell you pretty directly some of the most interesting things about the irreducible representations. So it would be great to understand the mu's. For (nontrivial) Weyl groups, there is an upper bound (# positive roots -1)/2 on degrees of KL polynomials; and this bound appears never to be attained except for SL(2). Part of the reason is that P_{p',p} is simpler when p has many descents, and the long element p=w_0 has LOTS of descents. But for a real group, the local system on the KGB orbit (indexed by the parameter lambda) can be complicated, and can prevent p from having many descents. Similarly, the parameter p' can have compact imaginary roots, which means it can have unexpectedly few ascents. When these two things combine, it can happen that P_{p',p} can have surprisingly high degree: close to or attaining the theoretical maximum of (#positive roots for G - #positive roots for K - 1)/2. Here's an example: G=Sp(4,R) with 4 positive roots, K=U(2) with 1 positive root, maximum degree of a KL polynomial (4-1-1)/2 = 1. atlas> set G=Sp(4,R) Variable G: RealForm atlas> set B=block_of(trivial(G)) Variable B: [Param] atlas> set M=KL_P_polynomials(trivial(G)) Variable M: [[vec]] atlas> printPolyMatrix (M) +1 0 0 0 +1 +1 0 +1 +1 +1 +1 0 0 +1 0 0 +1 0 +1 +1 +1 +1 +1 0 0 0 +1 0 0 +1 0 +1 0 +1 +1 +q 0 0 0 +1 0 0 +1 0 +1 +1 +1 +q 0 0 0 0 +1 0 0 +1 +1 +1 +1 0 0 0 0 0 0 +1 0 +1 0 +1 +1 0 0 0 0 0 0 0 +1 0 +1 +1 +1 0 0 0 0 0 0 0 0 +1 0 0 +1 0 0 0 0 0 0 0 0 0 +1 0 +1 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 0 0 0 0 +1 atlas> print_block(trivial(G)) Parameter defines element 10 of the following block: 0: 0 [i1,i1] 1 2 ( 4, *) ( 5, *) *(x= 0,lam_rho= [0,0], nu= [0,0]/1) e 1: 0 [i1,i1] 0 3 ( 4, *) ( 6, *) *(x= 1,lam_rho= [0,0], nu= [0,0]/1) e 2: 0 [ic,i1] 2 0 ( *, *) ( 5, *) *(x= 2,lam_rho= [0,0], nu= [0,0]/1) e 3: 0 [ic,i1] 3 1 ( *, *) ( 6, *) *(x= 3,lam_rho= [0,0], nu= [0,0]/1) e 4: 1 [r1,C+] 4 9 ( 0, 1) ( *, *) *(x= 4,lam_rho= [0,0], nu= [1,-1]/2) 1^e 5: 1 [C+,r1] 7 5 ( *, *) ( 0, 2) *(x= 5,lam_rho= [0,0], nu= [0,1]/1) 2^e 6: 1 [C+,r1] 8 6 ( *, *) ( 1, 3) *(x= 6,lam_rho= [0,0], nu= [0,1]/1) 2^e 7: 2 [C-,i1] 5 8 ( *, *) (10, *) *(x= 7,lam_rho= [0,0], nu= [2,0]/1) 1x2^e 8: 2 [C-,i1] 6 7 ( *, *) (10, *) *(x= 8,lam_rho= [0,0], nu= [2,0]/1) 1x2^e 9: 2 [i2,C-] 9 4 (10,11) ( *, *) *(x= 9,lam_rho= [0,0], nu= [3,3]/2) 2x1^e 10: 3 [r2,r1] 11 10 ( 9, *) ( 7, 8) *(x=10,lam_rho= [0,0], nu= [2,1]/1) 1^2x1^e 11: 3 [r2,rn] 10 11 ( 9, *) ( *, *) *(x=10,lam_rho= [1,1], nu= [2,1]/1) 1^2x1^e The print_block command describes the representations in the 12 rows (or columns) of the KL polynomial matrix. Notice that a degree 1 polynomial shows up exactly in entries (2,11) and (3,11). Sp(6,R) is clumsier. The maximum dimension difference is 6, corresponding to a maximum degree for KL polynomials of 2. The q^2 terms appear only in the upper right of the KL matrix: atlas> set G=Sp(6,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> print_block(trivial(G)) Parameter defines element 50 of the following block: 0: 0 [i1,i1,i1] 1 2 3 ( 8, *) (10, *) (12, *) *(x= 0,lam_rho= [0,0,0], nu= [0,0,0]/1) e 1: 0 [i1,ic,i1] 0 1 4 ( 8, *) ( *, *) (13, *) *(x= 1,lam_rho= [0,0,0], nu= [0,0,0]/1) e 2: 0 [ic,i1,i1] 2 0 5 ( *, *) (10, *) (14, *) *(x= 2,lam_rho= [0,0,0], nu= [0,0,0]/1) e 3: 0 [i1,ic,i1] 4 3 0 ( 9, *) ( *, *) (12, *) *(x= 3,lam_rho= [0,0,0], nu= [0,0,0]/1) e 4: 0 [i1,i1,i1] 3 6 1 ( 9, *) (11, *) (13, *) *(x= 4,lam_rho= [0,0,0], nu= [0,0,0]/1) e 5: 0 [ic,ic,i1] 5 5 2 ( *, *) ( *, *) (14, *) *(x= 5,lam_rho= [0,0,0], nu= [0,0,0]/1) e 6: 0 [ic,i1,i1] 6 4 7 ( *, *) (11, *) (15, *) *(x= 6,lam_rho= [0,0,0], nu= [0,0,0]/1) e 7: 0 [ic,ic,i1] 7 7 6 ( *, *) ( *, *) (15, *) *(x= 7,lam_rho= [0,0,0], nu= [0,0,0]/1) e 8: 1 [r1,C+,i1] 8 17 9 ( 0, 1) ( *, *) (16, *) *(x= 8,lam_rho= [0,0,0], nu= [1,-1,0]/2) 1^e 9: 1 [r1,C+,i1] 9 18 8 ( 3, 4) ( *, *) (16, *) *(x= 9,lam_rho= [0,0,0], nu= [1,-1,0]/2) 1^e 10: 1 [C+,r1,C+] 17 10 23 ( *, *) ( 0, 2) ( *, *) *(x=10,lam_rho= [0,0,0], nu= [0,1,-1]/2) 2^e 11: 1 [C+,r1,C+] 18 11 24 ( *, *) ( 4, 6) ( *, *) *(x=11,lam_rho= [0,0,0], nu= [0,1,-1]/2) 2^e 12: 1 [i1,C+,r1] 13 19 12 (16, *) ( *, *) ( 0, 3) *(x=12,lam_rho= [0,0,0], nu= [0,0,1]/1) 3^e 13: 1 [i1,C+,r1] 12 20 13 (16, *) ( *, *) ( 1, 4) *(x=13,lam_rho= [0,0,0], nu= [0,0,1]/1) 3^e 14: 1 [ic,C+,r1] 14 21 14 ( *, *) ( *, *) ( 2, 5) *(x=14,lam_rho= [0,0,0], nu= [0,0,1]/1) 3^e 15: 1 [ic,C+,r1] 15 22 15 ( *, *) ( *, *) ( 6, 7) *(x=15,lam_rho= [0,0,0], nu= [0,0,1]/1) 3^e ... 44: 5 [r2,C-,i1] 45 37 46 (42, *) ( *, *) (50, *) *(x=40,lam_rho= [0,0,0], nu= [3,2,0]/1) 1^2x1x3x2^e 45: 5 [r2,C-,i1] 44 38 47 (42, *) ( *, *) (51, *) *(x=40,lam_rho= [1,1,0], nu= [3,2,0]/1) 1^2x1x3x2^e 46: 5 [r2,C-,i1] 47 39 44 (43, *) ( *, *) (50, *) *(x=41,lam_rho= [0,0,0], nu= [3,2,0]/1) 1^2x1x3x2^e 47: 5 [r2,C-,i1] 46 40 45 (43, *) ( *, *) (51, *) *(x=41,lam_rho= [1,1,0], nu= [3,2,0]/1) 1^2x1x3x2^e 48: 5 [C-,i2,C-] 41 48 36 ( *, *) (50,52) ( *, *) *(x=42,lam_rho= [0,0,0], nu= [6,3,3]/2) 1x3x2x1^3^e 49: 5 [i2,C-,r1] 49 41 49 (50,51) ( *, *) (42,43) *(x=43,lam_rho= [0,0,0], nu= [5,5,2]/2) 2x3x2x1^3^e 50: 6 [r2,r2,r1] 51 52 50 (49, *) (48, *) (44,46) *(x=44,lam_rho= [0,0,0], nu= [3,2,1]/1) 1^2x3x2x1^3^e 51: 6 [r2,rn,r1] 50 51 51 (49, *) ( *, *) (45,47) *(x=44,lam_rho= [1,1,0], nu= [3,2,1]/1) 1^2x3x2x1^3^e 52: 6 [rn,r2,rn] 52 50 52 ( *, *) (48, *) ( *, *) *(x=44,lam_rho= [0,1,1], nu= [3,2,1]/1) 1^2x3x2x1^3^e atlas> printPolyMatrix (KL_P_polynomials(trivial(G))) {skip row 0; what follows is row 1, with its 54 entries arranged in seven lines} 0 +1 0 0 0 0 0 0 +1 0 0 0 0 +1 0 0 +1 +1 0 0 +1 0 0 0 +1 +1+q 0 0 +1 +q +1 +1 +1 0 +1 +1 +1 +1 0 +1 +q +1+q +1 +1+q +1 0 +1 +q +1+q +1 +1 +q +q+q^2 {skip row 2; row 3 is} 0 0 0 +1 0 0 0 0 0 +1 0 0 +1 0 0 0 +1 0 +1 +1 0 0 0 +1 0 +1+q +1 +q 0 0 +1 +1 0 +1 +1 +1 +1 +1 +q +1 0 +1+q +1+q +1 +1 +q +1 0 +1+q +1 +1 +q +q+q^2 {skip row 4; row 5 is} 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 0 +1 0 0 +1 +q 0 0 0 0 +1 0 +1 0 0 +1+q +q 0 0 +1 +1+q 0 +1+q^2 +q 0 0 +1 +1 +1 +q +q [skip row 6] 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 0 +1 0 0 0 +1 +q 0 0 0 +1 0 +1 0 0 0 +1+q +q +1 0 +1+q 0 0 +1+q^2 +q +1 +1 +1 +q +q This all the rows having q^2: rows 1,3,5,7; and the q^2 appears in columns 52, 44, 46. I would say that the _interesting_ entries are 1+q^2 in (5,44) and (7,46), where the dimension difference is precisely 5. So the problem is this: find some interesting pairs (p',p) of parameters with length(p)-length(p')=2d+1, and P_{p',p} of degree d In the quasisplit equal rank form of E6 there is a pair of parameters (p',p) with length difference 19; p' is a discrete series, and p attached to a codimension one orbit. The polynomial P_{p',p} is q^9+1. ************************* Second problem is about constant terms of P_{p',p}. In the Weyl group case KL proved that the constant term is either 1 (if p' \le p in the Bruhat order) or 0. In the group case this is not true: constant terms of 0 and any power of 2 can arise if p'\le p. I'll try to show some examples of each of these things. The problems are a) find a nice combinatorial description of the condition constant term(P_{p',p} = 0), and prove that a nonzero constant term is ALWAYS a power of 2. The software may be helpful in producing examples of the first phenomenon, which could lead to a conjecture. ************************** Third problem is for G=U(p,q): if X is any irreducible HC module of infinitesimal character belonging to (Z+e/2)^n (with p+q=2m+e), AND X is unitary, THEN X is a weakly fair A_q(lambda). ************************** notes on video 3:00 recall KL polys P_{p',p} for Harish-Chandra modules 5:00 started sharing WRONG damned screen. I'm very sorry about that! Here is approximately what I typed. atlas> set G=SL(2,R) Variable G: RealForm atlas> set p=trivial(G) Variable p: Param atlas> set M=KL_P_polynomials(p) Variable M: [[vec]] {these are Timothy's KL polynomials} atlas> M Value: [[[ 1 ],[ ],[ 1 ]],[[ ],[ 1 ],[ 1 ]],[[ ],[ ],[ 1 ]]] atlas> printPolyMatrix (M) +1 0 +1 0 +1 +1 0 0 +1 {here's how to get explanations of rows/columns of M: one for each block element} atlas> print_block(p) Parameter defines element 2 of the following block: 0: 0 [i1] 1 (2,*) *(x=0,lam_rho= [0], nu= [0]/1) e 1: 0 [i1] 0 (2,*) *(x=1,lam_rho= [0], nu= [0]/1) e 2: 1 [r1] 2 (0,1) *(x=2,lam_rho= [0], nu= [1]/1) 1^e {First two entries are discrete series, living on closed K-orbits=points in G/B; last is trivial rep, living on open orbit} {KL matrix is _formulas_ for irreps in terms of standards} {Timothy: degree(P_{p',p}) \le (length(p) - length(p'))/2} {\le is < if p' \ne p. SL(2): degree bound is 1/2, so all P are CONSTANT} {Now 14:00} atlas> set G=Sp(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> M:=KL_P_polynomials(trivial(G)) Value: [[[ 1 ],[ ],[ ],[ ],[ 1 ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ ]],[[ ],[ 1 ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ ]],[[ ],[ ],[ 1 ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 0, 1 ]],[[ ],[ ],[ ],[ 1 ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 0, 1 ]],[[ ],[ ],[ ],[ ],[ 1 ],[ ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ],[ ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ]]] atlas> printPolyMatrix (M) +1 0 0 0 +1 +1 0 +1 +1 +1 +1 0 0 +1 0 0 +1 0 +1 +1 +1 +1 +1 0 0 0 +1 0 0 +1 0 +1 0 +1 +1 +q 0 0 0 +1 0 0 +1 0 +1 +1 +1 +q 0 0 0 0 +1 0 0 +1 +1 +1 +1 0 0 0 0 0 0 +1 0 +1 0 +1 +1 0 0 0 0 0 0 0 +1 0 +1 +1 +1 0 0 0 0 0 0 0 0 +1 0 0 +1 0 0 0 0 0 0 0 0 0 +1 0 +1 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 0 0 0 0 +1 atlas> print_block(trivial(G)) Parameter defines element 10 of the following block: 0: 0 [i1,i1] 1 2 ( 4, *) ( 5, *) *(x= 0,lam_rho= [0,0], nu= [0,0]/1) e 1: 0 [i1,i1] 0 3 ( 4, *) ( 6, *) *(x= 1,lam_rho= [0,0], nu= [0,0]/1) e 2: 0 [ic,i1] 2 0 ( *, *) ( 5, *) *(x= 2,lam_rho= [0,0], nu= [0,0]/1) e 3: 0 [ic,i1] 3 1 ( *, *) ( 6, *) *(x= 3,lam_rho= [0,0], nu= [0,0]/1) e 4: 1 [r1,C+] 4 9 ( 0, 1) ( *, *) *(x= 4,lam_rho= [0,0], nu= [1,-1]/2) 1^e 5: 1 [C+,r1] 7 5 ( *, *) ( 0, 2) *(x= 5,lam_rho= [0,0], nu= [0,1]/1) 2^e 6: 1 [C+,r1] 8 6 ( *, *) ( 1, 3) *(x= 6,lam_rho= [0,0], nu= [0,1]/1) 2^e 7: 2 [C-,i1] 5 8 ( *, *) (10, *) *(x= 7,lam_rho= [0,0], nu= [2,0]/1) 1x2^e 8: 2 [C-,i1] 6 7 ( *, *) (10, *) *(x= 8,lam_rho= [0,0], nu= [2,0]/1) 1x2^e 9: 2 [i2,C-] 9 4 (10,11) ( *, *) *(x= 9,lam_rho= [0,0], nu= [3,3]/2) 2x1^e 10: 3 [r2,r1] 11 10 ( 9, *) ( 7, 8) *(x=10,lam_rho= [0,0], nu= [2,1]/1) 1^2x1^e 11: 3 [r2,rn] 10 11 ( 9, *) ( *, *) *(x=10,lam_rho= [1,1], nu= [2,1]/1) 1^2x1^e {P_{0 to 3, 10 to 11} gives degree bound of (3-1)/2 = 1, so could have LINEAR KL poly!} {This is last two columns, first four rows} {Two of these eight entries ARE nonconstant} {about 19:00} atlas> G:=Sp(4,C) Value: connected quasisplit real group with Lie algebra 'sp(4,C)' atlas> p:=trivial(G) Value: final parameter(x=7,lambda=[2,1,2,1]/1,nu=[2,1,2,1]/1) atlas> M:=KL_P_polynomials(p) Value: [[[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ ],[ ],[ 1 ],[ ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ ],[ ],[ ],[ 1 ],[ 1 ],[ 1 ],[ 1 ]],[[ ],[ ],[ ],[ ],[ ],[ 1 ],[ ],[ 1 ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ],[ 1 ]],[[ ],[ ],[ ],[ ],[ ],[ ],[ ],[ 1 ]]] atlas> printPolyMatrix (M) +1 +1 +1 +1 +1 +1 +1 +1 0 +1 0 +1 +1 +1 +1 +1 0 0 +1 +1 +1 +1 +1 +1 0 0 0 +1 0 +1 +1 +1 0 0 0 0 +1 +1 +1 +1 0 0 0 0 0 +1 0 +1 0 0 0 0 0 0 +1 +1 0 0 0 0 0 0 0 +1 atlas> print_block(p) Parameter defines element 7 of the following block: 0: 0 [C+,C+,C+,C+] 1 2 1 2 (*,*) (*,*) (*,*) (*,*) *(x=0,lam_rho= [0,0,0,0], nu= [0,0,0,0]/1) e 1: 1 [C-,C+,C-,C+] 0 4 0 3 (*,*) (*,*) (*,*) (*,*) *(x=1,lam_rho= [0,0,0,0], nu= [1,-1,1,-1]/2) 1xe 2: 1 [C+,C-,C+,C-] 3 0 4 0 (*,*) (*,*) (*,*) (*,*) *(x=2,lam_rho= [0,0,0,0], nu= [0,1,0,1]/1) 2xe 3: 2 [C-,C+,C+,C-] 2 6 5 1 (*,*) (*,*) (*,*) (*,*) *(x=3,lam_rho= [0,0,0,0], nu= [3,-1,1,3]/2) 1x2xe 4: 2 [C+,C-,C-,C+] 5 1 2 6 (*,*) (*,*) (*,*) (*,*) *(x=4,lam_rho= [0,0,0,0], nu= [1,3,3,-1]/2) 2x1xe 5: 3 [C-,C+,C-,C+] 4 7 3 7 (*,*) (*,*) (*,*) (*,*) *(x=5,lam_rho= [0,0,0,0], nu= [2,0,2,0]/1) 1x2x1xe 6: 3 [C+,C-,C+,C-] 7 3 7 4 (*,*) (*,*) (*,*) (*,*) *(x=6,lam_rho= [0,0,0,0], nu= [3,3,3,3]/2) 2x1x2xe 7: 4 [C-,C-,C-,C-] 6 5 6 5 (*,*) (*,*) (*,*) (*,*) *(x=7,lam_rho= [0,0,0,0], nu= [2,1,2,1]/1) 1x2x1x2 {suggests: look for _linear_ KL polys in upper right of matrix (length differences at least} {three: first three rows, last three columns. Those nine entries are all 1, not linear.} {SO real groups are easier/better than complex groups for producing interesting KL polys} atlas> G:=E7_s Value: connected split real group with Lie algebra 'e7(R)' atlas> atlas> set B=block_of(trivial(G)) Variable B: [Param] atlas> #B Value: 24678 atlas> length(B[24677]) Value: 35 atlas> length(B[0]) Value: 0 {SO degree bound for KL polys is 17} {Suggests: interesting to look for KL polys for E7_s of degree 17} {In video, computed them using Fokko. I'll do that, then repeat it in atlas as} {Jeff suggested} [dav@MATHSTATION040 atlassem]$ Fokko This is 'Fokko', interface to the Atlas of Lie Groups and Representations. Library version 1.0.7, build date: Oct 8 2019 at 09:19:46. Enter "help" if you need assistance. empty: kllist Lie type: E7 torsion subgroup of the center of the simply connected group is: Z/2 enter kernel generators, one per line (ad for adjoint, sc for simply connected, ? to abort): sc enter inner class(es): s (weak) real forms are: 0: e7 1: e7(e6.u(1)) 2: e7(so(12).su(2)) 3: e7(R) enter your choice: 3 possible (weak) dual real forms are: 0: e7 1: e7(e6.u(1)) 2: e7(so(12).su(2)) 3: e7(R) enter your choice: 3 computing Kazhdan-Lusztig polynomials ... l= 0, y= 71, polys: 2, mat: 0 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 1, y= 211, polys: 2, mat: 1708 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 2, y= 421, polys: 2, mat: 5708 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 3, y= 708, polys: 3, mat: 17172 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 4, y= 1084, polys: 4, mat: 41350 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 5, y= 1563, polys: 8, mat: 87068 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 6, y= 2144, polys: 11, mat: 165414 CPU time = 0 secs, Max res size= 71MB, pmem= 0MB, matmem= 0MB l= 7, y= 2833, polys: 21, mat: 287474 CPU time = 0 secs, Max res size= 72MB, pmem= 0MB, matmem= 1MB l= 8, y= 3627, polys: 42, mat: 474554 CPU time = 0 secs, Max res size= 72MB, pmem= 0MB, matmem= 1MB l= 9, y= 4527, polys: 106, mat: 748418 CPU time = 0 secs, Max res size= 74MB, pmem= 0MB, matmem= 2MB l= 10, y= 5523, polys: 214, mat: 1123772 CPU time = 0 secs, Max res size= 75MB, pmem= 0MB, matmem= 4MB l= 11, y= 6608, polys: 472, mat: 1608202 CPU time = 0 secs, Max res size= 77MB, pmem= 0MB, matmem= 6MB l= 12, y= 7766, polys: 1050, mat: 2234414 CPU time = 0 secs, Max res size= 80MB, pmem= 0MB, matmem= 8MB l= 13, y= 8983, polys: 2301, mat: 3006622 CPU time = 0 secs, Max res size= 83MB, pmem= 0MB, matmem= 11MB l= 14, y= 10238, polys: 4483, mat: 3947915 CPU time = 0 secs, Max res size= 87MB, pmem= 0MB, matmem= 15MB l= 15, y= 11511, polys: 9119, mat: 5007225 CPU time = 1 secs, Max res size= 92MB, pmem= 0MB, matmem= 19MB l= 16, y= 12793, polys: 17367, mat: 6215179 CPU time = 1 secs, Max res size= 97MB, pmem= 1MB, matmem= 23MB l= 17, y= 14069, polys: 31165, mat: 7564079 CPU time = 1 secs, Max res size= 104MB, pmem= 1MB, matmem= 28MB l= 18, y= 15316, polys: 51334, mat: 9017596 CPU time = 2 secs, Max res size= 111MB, pmem= 2MB, matmem= 34MB l= 19, y= 16506, polys: 77582, mat: 10508935 CPU time = 2 secs, Max res size= 119MB, pmem= 5MB, matmem= 40MB l= 20, y= 17633, polys: 116808, mat: 12053079 CPU time = 2 secs, Max res size= 128MB, pmem= 6MB, matmem= 45MB l= 21, y= 18685, polys: 161574, mat: 13560990 CPU time = 3 secs, Max res size= 137MB, pmem= 11MB, matmem= 51MB l= 22, y= 19658, polys: 222194, mat: 15061428 CPU time = 3 secs, Max res size= 148MB, pmem= 13MB, matmem= 57MB l= 23, y= 20537, polys: 291503, mat: 16460008 CPU time = 4 secs, Max res size= 159MB, pmem= 21MB, matmem= 62MB l= 24, y= 21324, polys: 369027, mat: 17758680 CPU time = 4 secs, Max res size= 168MB, pmem= 23MB, matmem= 67MB l= 25, y= 22011, polys: 453511, mat: 18922153 CPU time = 5 secs, Max res size= 181MB, pmem= 28MB, matmem= 72MB l= 26, y= 22605, polys: 546252, mat: 19973218 CPU time = 5 secs, Max res size= 201MB, pmem= 42MB, matmem= 76MB l= 27, y= 23104, polys: 630792, mat: 20829240 CPU time = 6 secs, Max res size= 201MB, pmem= 45MB, matmem= 79MB l= 28, y= 23519, polys: 717183, mat: 21572416 CPU time = 6 secs, Max res size= 209MB, pmem= 48MB, matmem= 82MB l= 29, y= 23853, polys: 791873, mat: 22151592 CPU time = 6 secs, Max res size= 217MB, pmem= 50MB, matmem= 84MB l= 30, y= 24118, polys: 856337, mat: 22621826 CPU time = 7 secs, Max res size= 230MB, pmem= 56MB, matmem= 86MB l= 31, y= 24318, polys: 904706, mat: 22959601 CPU time = 7 secs, Max res size= 232MB, pmem= 58MB, matmem= 87MB l= 32, y= 24466, polys: 940753, mat: 23210633 CPU time = 7 secs, Max res size= 235MB, pmem= 59MB, matmem= 88MB l= 33, y= 24571, polys: 962786, mat: 23368406 CPU time = 7 secs, Max res size= 237MB, pmem= 60MB, matmem= 89MB l= 34, y= 24641, polys: 978677, mat: 23480702 CPU time = 7 secs, Max res size= 239MB, pmem= 60MB, matmem= 89MB l= 35, y= 24677, polys: 993490, mat: 23568347 CPU time = 7 secs, Max res size= 240MB, pmem= 61MB, matmem= 89MB Total elapsed time = 8s. 993490 polynomials, 23568347 matrix entries. done Name an output file (return for stdout, ? to abandon): E7kllist block: qq [dav@MATHSTATION040 atlassem]$ tail E7kllist 4q^16+24q^15+116q^14+364q^13+718q^12+965q^11+916q^10+631q^9+315q^8+110q^7+25q^6+3q^5 4q^16+25q^15+125q^14+419q^13+867q^12+1170q^11+1094q^10+732q^9+343q^8+113q^7+25q^6+3q^5 4q^16+27q^15+123q^14+349q^13+659q^12+879q^11+840q^10+599q^9+308q^8+109q^7+25q^6+3q^5 4q^16+32q^15+156q^14+434q^13+803q^12+1024q^11+932q^10+633q^9+315q^8+110q^7+25q^6+3q^5 q^17+6q^16+22q^15+55q^14+110q^13+174q^12+226q^11+245q^10+214q^9+144q^8+72q^7+27q^6+6q^5+q^4 q^17+7q^16+26q^15+57q^14+88q^13+90q^12+60q^11+28q^10+7q^9+q^7 q^17+7q^16+27q^15+60q^14+90q^13+92q^12+62q^11+27q^10+7q^9+q^7 q^17+16q^16+62q^15+167q^14+314q^13+464q^12+533q^11+478q^10+344q^9+194q^8+80q^7+21q^6+3q^5 3q^17+16q^16+51q^15+126q^14+254q^13+406q^12+512q^11+507q^10+390q^9+229q^8+99q^7+31q^6+6q^5+q^4 3q^17+20q^16+82q^15+257q^14+560q^13+892q^12+1064q^11+944q^10+634q^9+315q^8+110q^7+25q^6+3q^5 {Notice six polynomials of degree 17! They don't look so simple/comprehensible, but that's why you're mathematicians; you figure stuff out!} {Here's the corresponding E6 example} [dav@MATHSTATION040 atlassem]$ tail E6kllist 2q^7+5q^6+12q^5+7q^4+2q^3 3q^7+3q^6+2q^5 3q^7+4q^6+2q^5 3q^7+7q^6+16q^5+7q^4+2q^3 3q^7+8q^6+11q^5+7q^4+2q^3 q^8+q^5 q^8+3q^7+9q^6+11q^5+7q^4+2q^3 q^8+4q^7+9q^6+12q^5+7q^4+2q^3 q^9+q^6 q^9+q^7 {Here's the F4 example:} q^5+q^4 q^5+q^4+q^2 q^5+2q^4+q^3+q^2 q^5+2q^4+3q^3+2q^2 q^5+2q^4+4q^3+3q^2+2q q^5+3q^4+q^3+q 2q^5+3q^4+q^2 2q^5+3q^4+2q^3+2q^2 2q^5+3q^4+3q^3+2q^2 q^6+q^5 {video 35:00} {Beginning of E6kllist:} 1 2 4 q q+1 q+2 q+4 2q 2q+1 2q+2 {Kazhdan-Lusztig proved only nonzero constant terms for W are all 1} {video 37:00} {Conjecture (about 2005): constant term of P_{p',p} is always 0 or power of 2} Atlas example: atlas> set G=SL(4,R) Variable G: RealForm atlas> set M=KL_P_polynomials(trivial(G)) Variable M: [[vec]] atlas> printPolyMatrix (M) +1 0 +1 +1 0 +1 +1 +1 +1 +1 +1 +1 +1 +q +q 0 +1 +1 0 +1 +1 +1 +1 +1 +1 +1 +1 +1 +q +q 0 0 +1 0 0 +1 +1 +1 +1 +1 +1 +1 +1 0 +q 0 0 0 +1 0 +1 +1 +1 0 +1 +1 +1 +1 +q 0 0 0 0 0 +1 +1 +1 0 +1 +1 +1 +1 +1 +q 0 0 0 0 0 0 +1 0 0 0 +1 +1 0 +1 0 0 0 0 0 0 0 0 +1 0 0 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 +1 0 +1 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 +1 +1 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 0 +1 0 0 +1 +1 0 0 0 0 0 0 0 0 0 0 0 +1 0 +1 0 +1 0 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 {biggest length is 4, so degree bound is 1, and nonconstant P actually appear!} {look at next to last column M[*][12]. Entry (2,13) is zero. But M[2][5]=1, so} {2 < 5. But also M[5][9] = 1, so 5<9; and finally M[9][13] = 1, so 9 < 13. So} {2 < 13 by transitivity, even though M[2][13]=0.} {Problem: use atlas to find examples of P_{p',p} = 0 even when p' < p in Bruhat order,} {and arrive at a geometric/combinatorial description of this situation.} {Use your solution to previous problem as a tool in proving Conjecture above.} {Misstatement in video: I _don't_ know whether atlas knows how to write down the Bruhat order on parameters, which is the transitive closure of the relation} { p' \le p whenever P_{p',p} \ne 0 .} {If it really doesn't exist as a script, then writing such a script is a great idea.} {atlas _does_ know the (compatible) closure relation on KGB, which is implemented by bruhat, bruhat_leq, and bruhat_geq} {video about 49:00 : atlas KL polys for F4...} atlas> set G=F4_s Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set p=trivial(G) Variable p: Param atlas> set M=KL_P_polynomials(p) {Wrote stupid code to print ALL the KL polynomials for F4:} for i:336 do for j:336 do printPoly(M[i][j]) od od {One thing that makes this horrible is that it prints () after each loop, which is to say} {336^2 times. Get rid of that by inserting void: at the beginning:} void: for i:336 do for j:336 do printPoly(M[i][j]) od od {This prints (just) 336^2 polynomials. Let's look only at those of degree six:} atlas> void: for i:336 do for j:336 do if(degree(M[i][j]) >= 5) then printPoly(M[i][j]) fi od od +q^4+q^5 +q^5 +2q^2+3q^3+2q^4+q^5 +q^4+q^5 +q^2+3q^4+2q^5 +q^5+q^6 +2q^2+3q^3+3q^4+2q^5 +q^5 +q+q^3+3q^4+q^5 +q^2+3q^4+2q^5 +2q+3q^2+4q^3+2q^4+q^5 +2q^2+3q^3+2q^4+q^5 +q^2+q^4+q^5 +2q^2+2q^3+3q^4+2q^5 +q^5 +1+2q^2+2q^3+q^5 +q^2+q^3+2q^4+q^5 +q^5 +q^4+q^5 +q^5 +q^2+q^4+q^5 +q^2+3q^4+2q^5 +2q^2+3q^3+2q^4+q^5 +q^2+q^3+2q^4+q^5 +q^5 +q^2+q^4+q^5 {Now make it tell us which i and j} atlas> void: for i:336 do for j:336 do if(degree(M[i][j]) =6) then prints("i = ",i, ", j = ",j, printPoly(M[i][j]),new_line) fi od od +q^5+q^6 i = 3, j = 335() {I don't understand why the polynomial appears first}