Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Leading
coeﬃcients of Kazhdan–Lusztig polynomials in type D Tyson Gern Department of Mathematics Thesis Defense March 21, 2013 Advisor: Dr. Richard M. Green

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Coxeter
groups and µ-values Let W be an arbitrary Coxeter group. In 1979 Kazhdan and Lusztig defined polynomials Px,w indexed by elements x, w ∈ W . Kazhdan and Lusztig conjectured that the coefficients of Px,w are non-negative. This conjecture is known to be true if W is an (affine) Weyl group. Recent work by Elias and Williamson suggests that this conjecture is true for arbitrary Coxeter groups.

groups and µ-values These polynomials are very difficult to compute. For example, if x < w in the Bruhat order, then we have deg(Px,w ) ≤ 1 2 ( (w) − (x) − 1), but it is unknown when this bound is reached. Computing the coefficient of the highest possible term, µ(x, w), of Px,w helps us to recursively compute Kazhdan–Lusztig polynomials. Unfortunately, the computation of these µ-values is not significantly easier than the computation of Px,w . There is no known algorithm that allows for the efficient computation of Px,w or µ(x, w), even in groups of relatively small rank.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result 0-1
Conjecture 0-1 Conjecture Let x, w ∈ Sn. Then µ(x, w) ∈ {0, 1}. This was disproved by McLarnan and Warrington in 2002. Using an improved recursive algorithm, they found the following counterexamples in S16 and S10 using computer calculations. Counterexample µ(54109832dc76bafe, c810d942fa53b6e7) = 5 µ(4321098765, 9467182350) = 4

Conjecture Computational evidence suggests that the 0-1 Conjecture holds in many cases. Known cases Let x, w ∈ Sn. We have µ(x, w) ∈ {0, 1} if one of the following holds: n ≤ 9 (McLarnan, Warrington); a(x) < a(w), where a is Lusztig’s a-function (Xi); there is at most one decreasing consecutive pair of entries in the one-line notation for w (Lascoux, Sch¨ utzenberger); x is fully commutative (Green).

Conjecture Theorem (Green) Let x, w ∈ W (An) such that x is fully commutative. Then µ(x, w) ∈ {0, 1}.

Conjecture Theorem (Green) Let x, w ∈ W (An) such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. We consider µ(x, w) where x is fully commutative in type D. Coxeter systems in type D are simply-laced. Type D has a nice representation as a subgroup of the signed permutation group.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Main
result Green’s proof in type A relies on the fact that descent sets in type A have certain favorable properties. In type D there are so-called bad elements which make µ-value computation much more diﬃcult. We classify all bad elements in type D and use domino tableaux and Lusztig’s a-function to compute their µ-values.

result Green’s proof in type A relies on the fact that descent sets in type A have certain favorable properties. In type D there are so-called bad elements which make µ-value computation much more diﬃcult. We classify all bad elements in type D and use domino tableaux and Lusztig’s a-function to compute their µ-values. Theorem (Gern) Let x, w ∈ W (Dn) be such that x is fully commutative. Then µ(x, w) ∈ {0, 1}.

system Deﬁnition A Coxeter system is a pair (W , S) consisting of a Coxeter group W and a set of generators S ⊂ W subject only to relations of the form (st)m(s,t) = 1, where m(s, s) = 1 and m(s, t) ≥ 2 for s = t in S.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Braid
relations Remark For each pair s, t ∈ S, the relation (st)m(s,t) = 1 can be rewritten as sts · · · m(s,t) = tst · · · m(s,t) , and is known as a braid relation. In particular, if m(s, t) = 2 then st = ts, so s and t commute. This is called a short braid relation. If m(s, t) ≥ 3 then we call the relation a long braid relation.

graph We encode the information of any Coxeter system (W , S) in a Coxeter graph Γ, where S is the vertex set for Γ. Whenever m(s, t) ≥ 3 we join the vertices s and t with an edge labeled by m(s, t), which we call the bond strength. As a convention, we omit the label when m(s, t) = 3. When all bond strengths are equal to 3 we call the system (W , S) simply-laced.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Type
An The Coxeter graph for W (An) is given by u u u u u q q q 1 2 3 n − 1 n The Coxeter group W (An) is generated by S(An) = {s1, s2, . . . , sn} subject to the following relations. s2 i = 1 for all i, si sj = sj si if |i − j| > 1, si sj si = sj si sj if |i − j| = 1. W (An) is isomorphic to the symmetric group Sn+1 under the isomorphism si → (i, i + 1).

Dn The Coxeter graph for W (Dn) is given by u u u u u u d d d q q q 1 2 3 4 n − 1 n . The Coxeter group W (Dn) is generated by S(Dn) = {s1, s2, . . . , sn} subject to the following relations. s2 i = 1 for all i, s1s2 = s2s1, s1s3s1 = s3s1s3, si sj = sj si if |i − j| > 1 and i, j ≥ 2, si sj si = sj si sj if |i − j| = 1 and i, j ≥ 2.

Dn The Coxeter graph for W (Dn) is given by u u u u u u d d d q q q 1 2 3 4 n − 1 n . W (Dn) is isomorphic to an index 2 subgroup of the signed permutation group Z2 Sn under the embedding si → (1, −2)(−1, 2) if i = 1; (i − 1, i)(−(i − 1), −i) if i ≥ 2.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Deﬁnition
Let I ⊂ S. Then the subgroup WI of W generated by I is called a parabolic subgroup of W . (WI , I) turns out to be a Coxeter system.

Let I ⊂ S. Then the subgroup WI of W generated by I is called a parabolic subgroup of W . (WI , I) turns out to be a Coxeter system. Example Let (W , S) be a Coxeter system of type Dn, and let I = {s2, s3, . . . , sn}. Then WI is a parabolic subgroup of W and (WI , I) is a Coxeter system of type An−1. u d d d 1 u u u u u q q q 2 3 4 n − 1 n .

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Reduced
expression Let (W , S) be a Coxeter system. Each w ∈ W may be written as a product w = si1 si2 · · · sir of generators from S. Deﬁnitions If r is minimal we call r the length of w, denoted (w). An expression of w as a product of (w) generators is called a reduced expression. A product w = v1v2 · · · vk is said to be reduced if (w) = k i=1 (vi ).

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Bruhat
order Definition Let W be a Coxeter group, let w ∈ W and let w = t1t2 · · · tk be a reduced expression for w. Then we can define a partial order on W as follows: x ≤ w ⇔ there exists a reduced expression x = ti1 ti2 · · · til , 1 ≤ i1 < i2 < · · · < il ≤ k. We call this partial order the Bruhat order. The choice of reduced expression for w does not matter, so the Bruhat order is indeed well-defined.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Descent
set Deﬁnition The right and left descent sets of an element w ∈ W are deﬁned by L(w) = {s ∈ S | (sw) < (w)} R(w) = {s ∈ S | (ws) < (w)}. It is known that s ∈ L(w) (R(w)) if and only if w has a reduced expression beginning (ending) in s.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Fully
commutative Let (W , S) be a simply laced Coxeter system and let w ∈ W . Deﬁnition We say that w is complex if w = w1 · sts · w2 reduced for some w1, w2 ∈ W and s, t ∈ S with m(s, t) = 3. Deﬁnition If an element w is not complex, then we call w fully commutative.

commutative Example Let W = W (D6), let x = s1s2s4s3s4 and let y = s1s2s6s3s5s4.

commutative Example Let W = W (D6), let x = s1s2s4s3s4 and let y = s1s2s6s3s5s4. Then x is not fully commutative since x = s1s2s4s3s4.

commutative Example Let W = W (D6), let x = s1s2s4s3s4 and let y = s1s2s6s3s5s4. Then x is not fully commutative since x = s1s2s4s3s4. However, y is fully commutative since we cannot apply any long braid relations to y.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Bad
elements Deﬁnition We call an element w of a Coxeter group W bad if 1 w is not a product of commuting generators, and 2 w has no reduced expressions beginning or ending in two non-commuting generators.

elements Deﬁnition We call an element w of a Coxeter group W bad if 1 w is not a product of commuting generators, and 2 w has no reduced expressions beginning or ending in two non-commuting generators. Theorem Coxeter groups of type A contain no bad elements.

elements Example Let W = W (D4), let x = s1s2s3s2s1, and let w = s1s2s4s3s1s2s4. Then

elements Example Let W = W (D4), let x = s1s2s3s2s1, and let w = s1s2s4s3s1s2s4. Then x = s1s2s3s2s1

elements Example Let W = W (D4), let x = s1s2s3s2s1, and let w = s1s2s4s3s1s2s4. Then x = s1s2s3s2s1 = s1s3s2s3s1,

elements Example Let W = W (D4), let x = s1s2s3s2s1, and let w = s1s2s4s3s1s2s4. Then x = s1s2s3s2s1 = s1s3s2s3s1, so x is not bad.

elements Example Let W = W (D4), let x = s1s2s3s2s1, and let w = s1s2s4s3s1s2s4. Then x = s1s2s3s2s1 = s1s3s2s3s1, so x is not bad. However, w has no reduced expressions beginning or ending in a pair of noncommuting generators, so w is bad.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Signed
permutations Recall that W (Dn) is isomorphic to an index 2 subgroup of the signed permutation group under the embedding si → (1, −2)(−1, 2) if i = 1; (i − 1, i)(−(i − 1), −i) if i ≥ 2. For an element σ ∈ Z2 Sn we will write σ using one-line notation: σ = (σ(1), σ(2), σ(3), . . . , σ(n)) .

permutations Recall that W (Dn) is isomorphic to an index 2 subgroup of the signed permutation group under the embedding si → (1, −2)(−1, 2) if i = 1; (i − 1, i)(−(i − 1), −i) if i ≥ 2. For an element σ ∈ Z2 Sn we will write σ using one-line notation: σ = (σ(1), σ(2), σ(3), . . . , σ(n)) . To simplify notation we underline negative numbers. For example, we write (2, −3, 4, −1) as (2, 3, 4, 1).

permutations In type D we can view descent sets in terms of signed permutations. Remark Let w ∈ W (Dn). Then R(w) = {si ∈ S | w(i − 1) > w(i)} and L(w) = si ∈ S | w−1(i − 1) > w−1(i) , where w(0) def = −w(2). This allows us to classify bad elements in type D using pattern avoidance.

signed permutations Theorem (Gern) Let W = W (Dn), n ≥ 4. W contains a unique longest bad element, wn. The only other bad elements in W are of the form wi · u reduced, where i ≤ n and u is a product of commuting generators. Furthermore, if n is odd then wn = wn−1. As a signed permutation, we have wn = (−1)n/2, n, 3, n − 2, 5, . . . , 4, n − 1, 2 when n is even.

signed permutations Example We have w4 = (1, 4, 3, 2); w6 = (1, 6, 3, 4, 5, 2); w8 = (1, 8, 3, 6, 5, 4, 7, 2); w10 = (1, 10, 3, 8, 5, 6, 7, 4, 9, 2).

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Heaps
A heap is partially ordered set associated to a reduced expression of an element in a Coxeter group. We use heaps to help visualize an element of a Coxeter group.

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space.

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s4 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s2s4 s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s1s2s4 s1s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s3s1s2s4 s3 s1s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s4s3s1s2s4 s4 s3 s1s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s2s4s3s1s2s4 s2 s4 s3 s1s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s1s2s4s3s1s2s4 s1s2 s4 s3 s1s2 s4

Example Let W = W (D4) and consider w4 = s1s2s4s3s1s2s4. Each generator is symbolized as a brick with width just greater than 1. We drop the bricks in order on a ﬂat surface and allow generators s1 and s2 to occupy the same space. w4 = s1s2s4s3s1s2s4. s1s2 s4 s3 s1s2 s4

To simplify notation, we often omit the boxes and use numbers to represent the generators of a Coxeter group when drawing heaps. s1s2 s4 s3 s1s2 s4 ⇒ 4 12 3 4 12

elements Example Here are a few examples of wn for small values of n. n 4 6 8 wn 12 2 3 12 4 12 4 6 3 5 12 4 3 5 12 4 6 12 4 6 8 3 5 7 12 4 6 3 5 12 4 6 12 4 6 8 3 5 7

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Example
Bad elements in W (D8): 12 4 3 12 4 12 4 6 3 12 4 12 4 8 3 12 4 12 4 7 3 12 4 12 4 6 8 3 12 4 12 4 6 3 5 12 4 3 5 12 4 6 12 4 6 8 3 5 12 4 3 5 12 4 6 12 4 6 8 3 5 7 12 4 6 3 5 12 4 6 12 4 6 8 3 5 7

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Hecke
Algebra Let A = Z[q 1 2 , q−1 2 ] be the ring of Laurent polynomials over Z. We deﬁne H(W , S) to be the algebra over A with linear basis {Tw | w ∈ W }, and multiplication determined by: TsTw = Tsw if (sw) > (w); qTsw + (q − 1)Tw if (sw) < (w).

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Hecke
Algebra Let A = Z[q 1 2 , q−1 2 ] be the ring of Laurent polynomials over Z. We deﬁne H(W , S) to be the algebra over A with linear basis {Tw | w ∈ W }, and multiplication determined by: TsTw = Tsw if (sw) > (w); qTsw + (q − 1)Tw if (sw) < (w). We deﬁne a ring homomorphism ι : H → H by ι(q) = q−1 and ι(Tw ) = T−1 w−1 . The map ι is an involution and gives rise to an interesting basis {Cw }w∈W for H.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Kazhdan–Lusztig
polynomials Theorem (Kazhdan, Lusztig) For each w ∈ W we have a unique element Cw ∈ H with the following properties: 1 ι(Cw ) = Cw , 2 Cw = x≤w (−1) (w)+ (x)q 1 2 ( (w)− (x))ι (Px,w ) Tx , where Pw,w = 1 and Px,w (q) ∈ Z[q] has degree ≤ 1 2 ( (w) − (x) − 1) if x < w.

polynomials Theorem (Kazhdan, Lusztig) For each w ∈ W we have a unique element Cw ∈ H with the following properties: 1 ι(Cw ) = Cw , 2 Cw = x≤w (−1) (w)+ (x)q 1 2 ( (w)− (x))ι (Px,w ) Tx , where Pw,w = 1 and Px,w (q) ∈ Z[q] has degree ≤ 1 2 ( (w) − (x) − 1) if x < w. The polynomials Px,w are called Kazhdan–Lusztig polynomials.

polynomials In general, Px,w is very diﬃcult to calculate. However, it is known that Px,w has the following properties: Pw,w = 1, Px,w = 0 if x > w, If x < w, then deg(Px,w ) ≤ 1 2 ( (w) − (x) − 1).

polynomials In general, Px,w is very diﬃcult to calculate. However, it is known that Px,w has the following properties: Pw,w = 1, Px,w = 0 if x > w, If x < w, then deg(Px,w ) ≤ 1 2 ( (w) − (x) − 1). The coeﬃcient of q 1 2 ( (w)− (x)−1) in Px,w is of particular interest, and is denoted µ(x, w).

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result µ-values
We know a few elementary properties of µ-values. µ(x, w) = 0 if x < w. If (w) − (x) is even, then µ(x, w) = 0 since 1 2 ( (w) − (x) − 1) ∈ Z. If s ∈ L(w) \ L(x) then either 1 x = sw and µ(x, w) = 1, or 2 x = sw and µ(x, w) = 0. If s ∈ R(w) \ R(x) then either 1 x = ws and µ(x, w) = 1, or 2 x = ws and µ(x, w) = 0. If x < w and µ(x, w) = 0 then we say x ≺ w.

Remark Understanding the µ values is a very important in computing Kazhdan–Lusztig polynomials. Amazingly, just understanding the leading coeﬃcients of these polynomials allows us to calculate them using the following recurrence relation: Px,w = q1−cPsx,sw + qcPx,sw − sz<z z≺sw µ(z, sw)q 1 2 ( (w)− (z))Px,z. where (sw) < (w) and c = 1, if (sx) < (x) 0, else .

Remark Understanding the µ values is a very important in computing Kazhdan–Lusztig polynomials. Amazingly, just understanding the leading coeﬃcients of these polynomials allows us to calculate them using the following recurrence relation: Px,w = q1−cPsx,sw + qcPx,sw − sz<z z≺sw µ(z, sw)q 1 2 ( (w)− (z))Px,z. where (sw) < (w) and c = 1, if (sx) < (x) 0, else . However, computing these µ-values is diﬃcult, even in cases of small rank.

An 0-1 Conjecture If W = W (An) then µ(x, w) ∈ {0, 1} for all x, w ∈ W

An 0-1 Conjecture If W = W (An) then µ(x, w) ∈ {0, 1} for all x, w ∈ W Theorem (Maclarnan, Warrington) The 0-1 Conjecture fails in groups of type An when n ≥ 9. However, even in groups of high rank, many of the µ-values are equal to 0 or 1.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Theorem
(Green) Let x, w ∈ W (An) such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. This proof relies on the fact that there are no bad elements in type A.

(Green) Let x, w ∈ W (An) such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. This proof relies on the fact that there are no bad elements in type A. Theorem (Gern) Let x, w ∈ W (Dn) be such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. To prove this we need to compute µ(xn, wn), where wn is bad and xn = s1s2s4s6 · · · sn.

(Green) Let x, w ∈ W (An) such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. This proof relies on the fact that there are no bad elements in type A. Theorem (Gern) Let x, w ∈ W (Dn) be such that x is fully commutative. Then µ(x, w) ∈ {0, 1}. To prove this we need to compute µ(xn, wn), where wn is bad and xn = s1s2s4s6 · · · sn. To do this, we will use Lusztig’s a-function.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Lusztig’s
a-function Let (W , S) be a Coxeter system. There is a function, a(w) : W → N, deﬁned by Lusztig that helps us to bound the degree of Pe,w , where w ∈ W and e is the identity element. deg(Pe,w ) ≤ 1 2 ( (w) − a(w))

a-function Let (W , S) be a Coxeter system. There is a function, a(w) : W → N, deﬁned by Lusztig that helps us to bound the degree of Pe,w , where w ∈ W and e is the identity element. deg(Pe,w ) ≤ 1 2 ( (w) − a(w)) This is of particular use to us, since it can be shown that Pe,wn = Pxn,wn , so deg(Pxn,wn ) ≤ 1 2 ( (wn) − a(wn))

a-function Lusztig’s a-function is defined in terms of structure constants in the Hecke algebra, and is very difficult to compute in general. If W is a finite Weyl group we have the following: If w = s1 · · · sr is made up of mutually commuting generators, then a(w) = r. If w0 is the longest element in W , then a(w0) = (w0). If I ⊂ S and w ∈ WI then a(w) calculated in terms of WI is the same as a(w) calculated in terms of W . There is a partition of W into sets called cells which behave nicely with respect to a.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Cells
Kazhdan and Lusztig deﬁned partitions of a Coxeter group W arising from equivalence relations ∼L, ∼R, and ∼LR. The equivalence classes corresponding to the relations ∼L, ∼R, and ∼LR are called left, right, and two-sided Kazhdan–Lusztig cells, respectively. It follows from the deﬁnitions that each two sided cell is a union of left cells (respectively, right cells).

Let W be a finite Weyl group and let x, w ∈ W . If x ∼LR w then a(x) = a(w). Like the a-function, cells are defined in terms of structure constants of the Hecke algebra. Thus, they are fairly difficult to compute. However, we can use domino tableaux to calculate cells.

A domino diagram is a collection of horizontal and vertical 2 × 1 dominoes such that the outline of the dominoes is in the shape of a Young diagram.

If we label the dominoes of a domino diagram with natural numbers such that the numbers do not repeat, and increase weakly down columns and increase weakly from left to right then we call the result a domino tableau.

If we label the dominoes of a domino diagram with natural numbers such that the numbers do not repeat, and increase weakly down columns and increase weakly from left to right then we call the result a domino tableau. 1 2 3 4 5 6 If a domino tableau is labeled with numbers 1 through n then we say that it is a standard domino tableau.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Shuﬄing
Given a domino tableau J and x ∈ N we can use a technique called shuﬄing to add either a horizontal or vertical domino labeled x to J to create a new domino tableau, denoted J ← x if the domino is horizontal, J ← (−x) if the domino is vertical.

Given a domino tableau J and x ∈ N we can use a technique called shuﬄing to add either a horizontal or vertical domino labeled x to J to create a new domino tableau, denoted J ← x if the domino is horizontal, J ← (−x) if the domino is vertical. Suppose that we are trying to add a vertical (horizontal) domino with label x. If x is larger than all the labels in J, then we may add x to the end of the ﬁrst column (row) of J.

Given a domino tableau J and x ∈ N we can use a technique called shuffling to add either a horizontal or vertical domino labeled x to J to create a new domino tableau, denoted J ← x if the domino is horizontal, J ← (−x) if the domino is vertical. Suppose that we are trying to add a vertical (horizontal) domino with label x. If x is larger than all the labels in J, then we may add x to the end of the first column (row) of J. If not, then we first remove all dominoes with label greater than x, add x to the end of the first column (row), Then shuffle the rest of the dominoes back into J.

When shuﬄing a domino back into a domino tableau there are three cases that we must consider. Case 1 If the domino does not overlap the tableau, then we can put it in its original place.

Case 2 If both squares of the domino overlap the tableau, then we shuﬄe it to the end of the next column (row).

Case 3 If only one of the squares of the domino overlap the tableau, then we shuﬄe it into the tableau by twisting.

Case 3 If only one of the squares of the domino overlap the tableau, then we shuﬄe it into the tableau by twisting. Let’s look at an example.

Example Let J be given as below. Let’s compute J ← 2. 1 3 5 4 ← 2 1

Example Let J be given as below. Let’s compute J ← 2. 1 ← 2 3 5 4 1

Example Let J be given as below. Let’s compute J ← 2. 1 3 5 4 1 2

Example Let J be given as below. Let’s compute J ← 2. 1 1 2 3 4 5 It is not obvious, yet true, that if J is a domino tableau, then J ← (±x) is also a domino tableau.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Domino
correspondence There is a bijection similar to the Robinson-Schensted correspondence, due to D. Garﬁnkle, between the signed permutation groups and pairs of standard domino tableaux with the same shape.

correspondence There is a bijection similar to the Robinson-Schensted correspondence, due to D. Garfinkle, between the signed permutation groups and pairs of standard domino tableaux with the same shape. Given an element σ ∈ Z2 Sn we will use shuffling to define a domino, TR(σ): TR(σ) = (((← σ(1)) ← σ(2)) ← σ(3)) · · · ← σ(n).

correspondence Now deﬁne TL(σ) = TR(σ−1). Remarkably, TR(σ) and TL(σ) have the same shape. Then we have the following result.

correspondence Now deﬁne TL(σ) = TR(σ−1). Remarkably, TR(σ) and TL(σ) have the same shape. Then we have the following result. Theorem (Garﬁnkle) The map σ → (TL(σ), TR(σ)) is a bijection from Z2 Sn to pairs of standard domino tableaux with the same shape, consisting of n dominoes.

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ).

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). ← 4

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 4

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 4 2 ←

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 ← 4

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 4 2

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 4

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 4 5 ←

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 4 5

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 4 5 ← 1

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 2 4 5 1

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 4 5 ← 3 1 2

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). ← 3 4 5 1 2

correspondence Example Let σ = (4, 2, 5, 1, 3). We calculate TR(σ). 4 5 1 2 3

correspondence Example Then σ−1 = (4, 2, 5, 1, 3) and TL(σ) = TR(σ−1) is 1 2 3 4 5

and tableaux In type A we can use standard tableaux to compute cells.

and tableaux In type A we can use standard tableaux to compute cells. Theorem Let x, w ∈ W = W (An). Then x ∼L w if and only if they have the same left tableaux according to the Robinson-Schensted correspondence.

and tableaux In type A we can use standard tableaux to compute cells. Theorem Let x, w ∈ W = W (An). Then x ∼L w if and only if they have the same left tableaux according to the Robinson-Schensted correspondence. Unfortunately, the analogous theorem does not hold for domino tableaux in type D, but we have something close. Theorem (Garﬁnkle) Let x, w ∈ W = W (Dn). Then x ∼L w if and only if TL(x) and TL(w) are open cycle equivalent.

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2)

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) ← 1

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 ← 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8 ← 3

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 ← 3 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8 3

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8 3 ← 6

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8 8 3 ← 6

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 8 3 6

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 ← 5

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 ← 5 6 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 5

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 5 ← 4

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 5 ← 4 5

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 5 5 4

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 6 8 5 4

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 4 6 8 ← 7

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 4 6 ← 7 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 4 6 8 7 ← 2

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 ← 2 3 4 5 6 7 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 3 4 5 6 7 8 2

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 4 5 6 7 8 3 2

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 4 5 6 7 8 2

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 6 7 8 2 4

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 8 2 4 6

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 2 4 6 8

We calculate TR(w8). w8 = (1, 8, 3, 6, 5, 4, 7, 2) 1 3 5 7 2 2 4 6 8 Since w8 = w−1 8 we have TL(w8) = TR(w8).

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Cycles
We will now introduce the notion of moving a domino tableau, T, through a cycle, C, in order to create a new tableau, E(T, C) which is open cycle equivalent to T. First we must show how to pivot dominoes within a tableau.

We will now introduce the notion of moving a domino tableau, T, through a cycle, C, in order to create a new tableau, E(T, C) which is open cycle equivalent to T. First we must show how to pivot dominoes within a tableau. We assign a coordinate (i, j) to each square in the domino tableaux, starting with (1, 1) in the upper left, and call a square ﬁxed if i + j is even.

Example Consider, for example, the graph of TL(w8), below. 1 2 3 4 5 7 6 8

Example Consider, for example, the graph of TL(w8), below. 1 2 3 4 5 7 6 8 The shaded squares represent the ﬁxed squares.

Next, we use the following rules to pivot each domino in a tableau, T, about its ﬁxed square in order to obtain a new tableau, T , r > k r < k r k k k r k r > k r < k r k k k r k where r = 0 if r lies above or to the left of the tableau, and r = ∞ if r lies below or to the right of the tableau.

Example Using these rules we can create (TL(w8)) . 1 2 3 4 5 7 6 8 r1 r3 r5 r7 r2 r4 r6 r8

Example Using these rules we can create (TL(w8)) . 1 3 4 5 7 6 8 r1 r3 r5 r7 r4 r6 r8 2

Example Using these rules we can create (TL(w8)) . 3 4 5 7 6 8 r3 r5 r7 r4 r6 r8 1 2

Example Using these rules we can create (TL(w8)) . 4 5 7 6 8 r5 r7 r4 r6 r8 1 2 3

Example Using these rules we can create (TL(w8)) . 4 7 6 8 r7 r4 r6 r8 1 2 3 5

Example Using these rules we can create (TL(w8)) . 4 6 8 r4 r6 r8 1 2 3 5 7

Example Using these rules we can create (TL(w8)) . 6 8 r6 r8 1 2 3 5 7 4

Example Using these rules we can create (TL(w8)) . 8 r8 1 2 3 5 7 4 6

Example Using these rules we can create (TL(w8)) . 1 2 3 5 7 4 6 8

Now we are ready to deﬁne the cycles of a domino tableau, T. We generate an equivalence relation ∼ on {1, 2, 3, . . . , n} by i ∼ j if the domino with label i in T overlaps with the domino with label j in T .

Now we are ready to deﬁne the cycles of a domino tableau, T. We generate an equivalence relation ∼ on {1, 2, 3, . . . , n} by i ∼ j if the domino with label i in T overlaps with the domino with label j in T . The equivalence classes associated to ∼ are called cycles.

Now we are ready to deﬁne the cycles of a domino tableau, T. We generate an equivalence relation ∼ on {1, 2, 3, . . . , n} by i ∼ j if the domino with label i in T overlaps with the domino with label j in T . The equivalence classes associated to ∼ are called cycles. A cycle, C, is closed if each domino of T with a label from C completely overlaps with T . If C is not closed, then C is open.

1 2 3 4 5 7 6 8 1 2 3 5 7 4 6 8

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result The
set C = {1, 2, 3, 5, 7} is an open cycle of TL(w8). 1 2 3 5 7 4 6 8

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result The
set C = {1, 2, 3, 5, 7} is an open cycle of TL(w8). If we pivot each domino with a label from C in TL(w8), we get a new tableaux E(TL(w8), C). Then TL(w8) is open cycle equivalent to E(TL(w8), C). 4 6 8 1 2 3 5 7

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Now
if we compute TL(s7s8s7 · w6) we obtain TL(s7s8s7w6) = E(TL(w8), C). Then TL(w8) is open cycle equivalent to TL(s7s8s7 · w6), thus w8 ∼L s7s8s7 · w6.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Now
if we compute TL(s7s8s7 · w6) we obtain TL(s7s8s7w6) = E(TL(w8), C). Then TL(w8) is open cycle equivalent to TL(s7s8s7 · w6), thus w8 ∼L s7s8s7 · w6. We can use TR to show that s7s8s7 · w6 ∼R w4 · s7s8s7. Then we have w8 ∼LR w4 · s7s8s7.

of bad elements Using similar methods we can calculate w4 ∼LR s1s2s4, so a(w4) = a(s1s2s4) = 3. w6 ∼LR s1s2s5s6s5, so a(w6) = a(s1s2s5s6s5) = 5.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result a-values
of bad elements If n ≥ 8 we can use tableaux to show that wn ∼LR wn−4sn−1snsn−1, thus a(wn) = a(wn−4sn−1snsn−1). sn−1snsn−1 is the longest element in the parabolic subgroup generated by I = {sn−1, sn}.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result a-values
of bad elements If n ≥ 8 we can use tableaux to show that wn ∼LR wn−4sn−1snsn−1, thus a(wn) = a(wn−4sn−1snsn−1). sn−1snsn−1 is the longest element in the parabolic subgroup generated by I = {sn−1, sn}. This gives us a recursive way to compute a(wn), since we have a(wn) = a(wn−4) + a(sn−1snsn−1) = a(wn−4) + 3.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Bounding
deg(Pxn,wn ) Lemma If n ∈ N is even, then a(wn) =      3n 4 if n ≡ 0 mod 4; 3n + 2 4 if n ≡ 2 mod 4.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Results
Recall deg(Pxn,wn ) ≤ 1 2 ( (wn) − a(wn)). Then we have deg(Pxn,wn ) ≤        3n2 16 − n 4 if n ≡ 0 mod 4; 3n2 16 − n 4 − 1 4 if n ≡ 2 mod 4.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Results
Recall deg(Pxn,wn ) ≤ 1 2 ( (wn) − a(wn)). Then we have deg(Pxn,wn ) ≤        3n2 16 − n 4 if n ≡ 0 mod 4; 3n2 16 − n 4 − 1 4 if n ≡ 2 mod 4. The coeﬃcient of µ(xn, wn) in Pxn,wn is on the term of degree 1 2 ( (wn) − (xn) − 1) = 3n2 16 − n 8 − 1.

Result For n > 8 we have a(wn) large enough that deg(Pxn,wn ) < 1 2 ( (wn) − (xn) − 1). Then µ(xn, wn) = 0 for n > 8. We can calculate µ(x4, w4) = 0 using elementary means, and µ(x6, w6) = 1, µ(x8, w8) = 0 using computer calculations. These µ-values give us information about µ(x, w) when x is fully commutative and w is arbitrary.

Result For n > 8 we have a(wn) large enough that deg(Pxn,wn ) < 1 2 ( (wn) − (xn) − 1). Then µ(xn, wn) = 0 for n > 8. We can calculate µ(x4, w4) = 0 using elementary means, and µ(x6, w6) = 1, µ(x8, w8) = 0 using computer calculations. These µ-values give us information about µ(x, w) when x is fully commutative and w is arbitrary. Theorem (Gern) Let x, w ∈ W (Dn) be such that x is fully commutative. Then µ(x, w) ∈ {0, 1}.

Overview Coxeter groups Kazhdan–Lusztig theory Domino tableaux Main result Special
thanks I’d like to thank my advisor, Richard Green, for his time and patience. Thank you to the other members of my committee for reading my thesis and being here today.

Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D

Leading Coefficients of Kazhdan–Lusztig Polynomials in Type D

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