Spin Representation Theory of Symmetric Groups and Related Combinatorics Notes from a Reading Course in Fall 2018

Spin Representation Theory of Symmetric Groups and Related Combinatorics Notes from a reading course in Fall 2018

George H. Seelinger 1. Introduction (presented by Jinkui Wan) When discussing the representation theory of the symmetric group, one considers linear representations which are group homomoprhisms

Sn → GL(V ) In 1911, Schur started considering projective representations ∗ Sn → P GL(V ) = GL(V )/C leading to the projective representation theory of Sn. It turns out that this corresponds to the linear representation theory of an extension of Sn, denoted S˜ n and referred to as the double cover of the symmetric group, ﬁtting into the short exact sequence ˜ 1 → Z/2Z → Sn → Sn → 1 ˜ where, if Z/2Z = {1, z}, then z is central in Sn, which gives us that z = 1 or z = −1.

When z = 1, we have the representation theory of Sn. When, z = −1, we have the representation theory of the spin symmetric group algebra 2 * ti = 1 + − CSn = CSn/hz + 1i = t1, . . . , tn | titi+1ti = ti+1titi+1 titj = −tjti when |i − j| > 1 which is equipped with a Z/2Z-grading. So, when we discuss spin represen- − tations of Sn, we are discussing linear representations of CSn . Our program to establish these ideas is as follows.

Part I (1) Basics of associative superalgebras (2) Connection to Hecke-Clifford (or Sengeev) algebra, Hn (3) Split conjugacy classes in a finite supergroup (4) Characteristic map (5) Schur-Q functions (6) Schur-Sergeev duality (7) Seminormal form of irreducible representations Part II − (1) Centers of CSn (analog of Farahat-Higman theory for CSn) − (2) Coinvariant theory for CSn (3) Spin Kostka polynomials (4) Quantum deformation (in particular, Olshanki-Sergeev duality) 2. Generalities for Associative Superalgebras (presented by Jinkui Wan) 2.1. Definitions and Examples.

2.1. Definition. (a)A vector superspace (over C) is a Z/2Z-graded vector space V = V0 ⊕ V1, where elements of V0 are called even and elements of V1 are called odd. For v ∈ Vi, i ∈ Z/2Z, we say |v| = i. (b) If V is a vector superspace with dim V0 = m and dim V1 = n, we say the graded dimension of V is (m, n), denoted dimV = (m, n). (c)A superalgebra is a C-algebra A with a Z/2Z-grading A = A0 ⊕ A1 such that AiAj ⊆ Ai+j for all i, j ∈ Z/2Z. (d)A superalgebra ideal is a homogeneous ideal, that is, a subset I ⊆ A such that I = I0 ⊕ I1 = (I ∩ A0) ⊕ (I ∩ A1) as vector spaces and AiIj ⊆ Ii+j for all i, j ∈ Z/2Z. (e) A superalgebra that has no non-trivial ideals is called simple. (f)A superalgebra homomorphism θ : A → B is an even algebra homomorphism, that is, an algebra homomorphism sending Ai → Bi for all i ∈ Z/2Z. (g) Given superalgebras A and B, the tensor product A ⊗ B is a superalgebra with multiplication (a ⊗ b)(a0 ⊗ b0) = (−1)|a||b|aa0 ⊗ bb0 for homogeneous elements and extended by linearity. (h)A commutative superalgebra is one that is graded commutative, that is yx = (−1)|x||y|xy Thus, the supercommutator of a superalgebra is given by [x, y] = xy − (−1)|x||y|yx and the supercenter is given by Z(A) = {a ∈ A | [a, x] = 0 for all x ∈ A} which is diﬀerent than the center of an ungraded algebra. (i) Given a superalgebra A, we let |A| be the associative algebra where we forget the grading on A.

2.2. Example. (a) Let V = V (m|n), the vector superspace with dimV = (m, n). Then, EndC(V ) is a superalgebra and is isomorphic to the matrix superalgebra    a is an m × m matrix   a b b is an m × n matrix  M(m|n) := |  c d c is an n × m matrix   d is an n × n matrix 

3 or, in other words, M(m|n) consists of all m|n-block matrices and has dimM(m|n) = (m2 + n2, 2mn). Furthermore, M(m|n) is a simple superalgebra since |M(m|n)| is simple as a C-algebra. (b) Let V = V (n|n) and p ∈ EndC(V ) be an odd involution (that is, it sends Vi → Vi+1 for i ∈ Z/2Z). Then, we deﬁne |f| Q(V ) := {f ∈ EndC(V ) | fp = (−1) pf} = Q(V )0 ⊕ Q(V )1

Q(V ) is also a superalgebra. Moreover, if we pick a basis {v1, . . . , vn} 0 of V0 and let vi = p(vi) for 1 ≤ i ≤ n, we have that, with respect to 0 0 the basis {v1, . . . , vn, v1, . . . , vn}, Q(V ) is isomorphic to a b Q(n) := ∈ M(n|n) −b a and it is simple. (c) The Cliﬀord algebra C`n is the superalgebra generated by the odd elements c1, . . . , cn subject to the relations

Proof. For part (a), we note that ∼ EndC(V (m|n)) ⊗ EndC(V (k|l)) = EndC(V (mk + ml|mk + nl)) under the isomorphism sending f ⊗ g to the endomorphism of V (mk + ml|mk + nl) mapping v ⊗ w to (−1)|g||v|f(v) ⊗ g(w). For part (b), we have End(V (m|n)) ⊗ Q(V (k|k), p) =∼ Q(V (m|n) ⊗ V (k|k), id ⊗ p)

For (c), one explicitly checks that Q(1) ⊗ Q(1) =∼ M(1|1) and then inductively applies (a) and (b) above. ∼ 2.4. Corollary. Since C`m+n = C`m ⊗ C`n under the isomorphism sending generators c1, . . . , cn to c1 ⊗ 1, . . . , cn ⊗ 1 and cn+1, . . . , cn+m to 1 ⊗ c1,..., 1 ⊗ cm, we have the corollaries ∼ (a) C`1 = Q(1) under the isomorphism c1 7→ p(v1) ∼ ∼ ∼ ∼ (b) C`2 = M(1|1) since C`2 = C`1 ⊗ C`1 = Q(1) ⊗ Q(1) = M(1|1) ∼ k−1 k−1 (c) C`2k = M(2 |2 ) ∼ k−1 (d) C`2k−1 = Q(2 ) (e) and thus, C`n is simple by parts (c) and (d).

4 2.2. Classiﬁcation of Simple Superalgebras. 2.5. Theorem. There are two types of ﬁnite dimensional simple associative superalgebras over C: (a) M(m|n) (b) Q(n) 2.3. Wedderburn Theorem and Schur’s Lemma. 2.6. Definition. (a)A (super)module over a superalgebra A is a vector space M = M0 ⊕ M1 with a left action of A on M such that AiMj ⊆ Mi+j for all i, j ∈ Z/2Z. (b) A homomorphism between A-modules M and N is a linear map f : M → N such that f(am) = (−1)|f||a|af(m) for all a ∈ A, m ∈ M and

HomA(M,N) := HomA(M,N)0 ⊕ HomA(M,N)1

where f ∈ HomA(M,N)1 ⊆ HomC(M,N) is such that f(Mi) ⊆ Ni+1 for all i ∈ Z/2Z. 2.7. Definition. An A-module is said to be simple if it is nonzero and has no proper A-submodules. An A-module M is said to be semisimple if every A-submodule of M is a direct summand of M. 2.8. Theorem (Super Wedderburn Theorem). The following are equiv- alent for a finite dimensional superalgebra A. (a) Every A-module is semisimple (b) A is a finite direct sum of left simple superideals (c) A is a direct product of a finite number of simple algebras 2.9. Definition. Thus, we say a superalgebra A is semisimple if it satisfies one of the three conditions.

2.10. Example. (a) M(m|n) = I1 ⊕I2 ⊕· · ·⊕Im ⊕Im+1 ⊕· · ·⊕Im+n where Ik = M(m|n)Ek,k for 1 ≤ k ≤ m + n. (b) Q(n) = J1 ⊕ · · · ⊕ Jn where

Jk = Q(n)(Ek,k + En+k,n+k) Check this. ∼ ∼ (c) HomM(m|n)(Ik,Ik) = C and HomQ(n)(Jk,Jk) = C ⊕ Cp. Impor- Most likely de- tantly, the latter space is not 1-dimensional despite Jk being 1- pends on your dimensional! choice of basis and involution. 2.11. Corollary. A ﬁnite dimensional semisimple superalgebra A is isomorphic to m n ∼ M M A = M(ri|si) ⊕ Q(nj) i=1 j=1

5 where m = m(A) and q = q(A) are invariants of A. 2.12. Definition. A simple A-module V is said to be of type M (resp. type Q) if it is annihilated by all but one summand of the form M(ri|si) (resp. Q(nj)). 2.13. Corollary. (a) The number of non-isomorphic simple A- modules is given by m(A) + q(A) = dim(Z(|A|) ∩ A0). (b) The number of non-isomorphic simple A-modules of type Q is given by q(A) = dim(Z(|A|) ∩ A1). 2.14. Theorem (Schur’s Lemma). If M and L are simple A-modules, then  1 if M =∼ L of type M  dim HomA(M,L) = 2 if M =∼ L of type Q 0 otherwise 2.15. Remark. (a) A simple A-module M is of type M if and only if |M| is a simple |A|-module (b) A simple A-module M is of type Q if and only if |M| is a direct sum of two non-isomorphic simple |A|-modules.

3. Split Conjugacy Classses in a Finite Supergroup (presented by Jinkui Wan)

Throughout, let G be a ﬁnite group with index 2 subgroup G0 ≤ G.

3.1. Definition. (a) We say that the elements of G0 are even elements and the elements of G1 := G \ G0 are odd elements, (b) CG is a superalgebra, which we will denote C[G, G0] 3.2. Theorem (Super MAschke’s Theorem). C[G, G0] is semisimple −1 3.3. Proposition. (a) If g ∈ Gi, h ∈ G, then hgh ∈ Gi for all i ∈ Z/2Z. (b) The number of non-isomorphic simple C[G, G0]-modules is equal to the number of even conjugacy classes in G. (c) The number of non-isomorphic simple C[G, G0]-module of type Q is equal to the number of odd conjugacy classes in G.

Proof. We note that, by the usual Artin-Wedderburn theorem, CG decomposes into a direct sum of simple matrix algebras, each of which has a 1-dimensional center and can be indexed by a conjugacy class of G via c = P g where C is a conjugacy class of G. In fact, this shows in the i g∈Ci i classical theory that the number of conjugacy classes of G equal the number of irreducible representations. Since conjugacy classes are either even or odd by (a), which is left as an exercise, (b) follows because dim(Z(CG)∩C[G, G0]0) is equal to the number of non-isomorphic simple C[G, G0]-modules and (c) follow from the fact that dim(Z(CG) ∩ C[G, G0]1) gives the number of those of type Q.

6 Now, consider the following situation. Let G˜ be a group such that there exists an index 2 subgroup G˜0 ≤ G˜ and there exists a short exact sequence 1 → {1, z} → G˜ →θ G → 1 where z2 = 1 and z is central in G˜. Then, 3.4. Proposition. For C a conjugacy class of G, the preimage θ−1(C) = {g, gz | g ∈ C} ⊆ G˜ has that (a) θ−1(C) is a single conjugacy class in G˜ if g is conjugate to zg in G˜ or (b) θ−1(C) splits into two conjugacy classes in G˜ if there exists a g ∈ C such that g is not conjugate to zg. In this case, we call C split. 3.5. Definition. We set ˜− CG := C[G, G0]/hz + 1i ˜− ˜ ˜ and call a CG -module a spin C[G, G0]-module. 3.6. Proposition. (a) We have the isomorphism of superalgebras ˜ ˜ ∼ ˜− C[G, G0] = C[G, G0] ⊕ CG | {z } | {z } (z=1) (z=−1) ˜ ˜ (b) The number of non-isomorphic simple spin C[G, G0]-modules is equal to the number of even split conjugacy classes of G. ˜ ˜ (c) The number of non-isomorphic simple spin C[G, G0]-modules of type Q is equal to the number of odd split conjugacy classes of G. ˜ ˜ Proof. Part (a) follows from the semisimplicity of C[G, G0]. Now, (a) tells us that ˜− ˜ ˜ Z(|CG |) = {a ∈ Z(|C[G, G0]|) | za = −a} So let

D1, zD1,D2, zD2,...,Dr, zDr,Dr+1,...,Dr+s | {z } | {z } split non-split be the conjugacy classes of G˜ where r is the number of split conjugacy classes in G, Di ∩ zDi = ∅ for 1 ≤ i ≤ r and zDj = Dj for r + 1 ≤ j ≤ r + s. Then, ˜ ˜ ˜ Z(|CG|) ∩ CG0 = {a ∈ Z(|CG|) | a is even and za = −a} has basis di1 − zdi1 , di2 − zdi2 , . . . , dik − zdik for di even and Check this part ˜ ˜ ˜ Z(|CG|) ∩ CG1 = {a ∈ Z(|CG|) | a is odd and za = −a} has basis dj1 − zdj1 , dj2 − zdj2 , . . . , djk − zdj` for dj odd.

7 3.7. Example. We have

θn 1 → {1, z} → S˜ n → Sn → 1 where z ∈ S˜ n is even and central, the subgroup of index 2 is A˜n, and − ˜ CSn := CSn/hz + 1i is the spin symmetric group algebra.

3.8. Definition. Throughout the remainder of these notes, we deﬁne θn : S˜ n → Sn to be the double covering may above.

4. A Morita Superequivalence (presented by Jinkui Wan)

Since Sn acts on C`n via σ.ci = cσ(i), we can deﬁne the semidirect product C`n o CSn with multiplication (x, σ)(y, τ) = (xσ(y), στ)

4.1. Definition. We deﬁne the Hecke-Cliﬀord superalgebra as

H := C`n o CSn with the ci having odd parity and the sj having even parity. 4.2. Lemma. There exists a superalgebra isomoprhism ∼ − CSn ⊗ C`n → Hn ci 7→ ci 1 tj 7→ √ sj(cj − cj+1) −2 Check this last map. Recall that C`n is a simple superalgebra and it has a unique simple module Un. If n is even, Un is of type M and if n is odd, Un is of type Q. This leads us to deﬁne two functors

Fn := − ⊗ Un : CSn-Mod → Hn-Mod − Gn := HomC`n (Un, −): Hn-Mod → CSn -Mod 4.3. Lemma. [Kle05, Prop 13.2.2] ∼ ∼ (a) If n is even, then Fn ◦ Gn = id and Gn ◦ Fn = id. ∼ ∼ (b) If n is odd, then Fn ◦ Gn = id ⊕ π and Gn ◦ Fn = id ⊕ π where π(M)i = Mi+1 for all i ∈ Z/2Z.

Thus, because (Fn ◦ Gn)(M) = HomC`n (Un,M) ⊗ Un, we have a (su- − per)Morita equivalence between CSn ⊗ C`n and Hn.

8 5. A Double Cover B˜n (presented by Jinkui Wan) n Recall that Bn = Z2 o Sn. We deﬁne 5.1. Definition. 2 2 z = ai = 1, ∀1 ≤ i ≤ n Πn := z, a1, . . . , an | aiaj = zajai, i 6= j

Then, Sn acts on Πn via σ(z) = z and σ(ai) = aσ(i). This gives us the short exact sequence

1 → {1, z} → Πn o Sn → Bn → 1 ai → bi and so we deﬁne The ﬂow here is ˜ not great. 5.2. Definition. Let Bn be the supergroup on Πn oSn with the ai odd, z even, and σ ∈ Sn even. ˜ Since CBn/hz + 1i = Hn, we wish to understand conjugacy classes in Bn. We will do so by example. 5.3. Example. Consider

x = ((+ + + − + + + − +−), (1234)(567)(89)) ∈ B10

As an element of S10, (1234)(567)(89) has cycle type (4, 3, 2, 1), but we wish to assign a parity to each of these cycles. To do so, we look at the (+, −)-array in the ﬁrst coordinate and take the product of the entries corresponding to the cycle. So, (1234) gets cycle type + × + × + × − = − since those are entries 1, 2, 3, and 4 in the array. This gives the cycle type as a tuple of partitions ρ = (ρ+, ρ−) and so ρ(x) = ((3), (4, 2, 1)). Similarly, if

y = ((+ − − − + − − − +−), (1386)(279)(45)) ∈ B10 then the ﬁrst cycle has parity + × − × − × − = − since those are the 1, 3, 6, and 8 entries of the array. One can check that y has the same cycle type as x.

5.4. Lemma. Two elements of Bn are conjugate if and only if their cycle types are the same.

5.5. Corollary. The number of conjugacy classes in Bn is #{(ρ+, ρ−) | |ρ+| + |ρ−| = n}

Now, the conjugacy class Cρ+,ρ− is even if k is even for bi1 bi2 . . . bik σ ∈ | {zn } ∈Z2 Cρ+,ρ− . 5.6. Theorem (Read). [CW12, Theorem 3.31] + − (a) Even Cρ+,ρ− splits if and only if ρ ∈ OPn and ρ = ∅ + − − (b) Odd Cρ+,ρ− splits if and only if ρ = ∅ and ρ ∈ SPn

9 − where SPn is all partitions of n with strict parts and odd length. + 5.7. Definition. For α ∈ OPn, let Cα be the split conjugacy class in B˜n satisfying + −1 (a) Cα = θn (Cα,∅) + (b) There exists σ ∈ Cα such that σ ∈ Sn with cycle type α.

6. A ring structure on R− (presented by Jinkui Wan)

6.1. Definition. We give the following definition − (a) Let Rn := [Hn-Mod], the Grothendieck group of Hn-Mod. − L∞ − − (b) Let R := n=0 Rn where R0 = Z (c) Let R− = ⊗ R− Q Q Z (d) Let Hm,n be the subalgebra of Hm+n generated by C`m+n and Sm × ∼ Sn. Note that Hm,n = Hm ⊗ Hn as a superalgebra. (e) Given M ∈ Hm-Mod and N ∈ Hn-Mod, we define [M] · [N] := [IndHm+n M ⊗ N] Hm⊗Hn 6.2. Proposition. R− is commutative with respect to the above multiplication. 6.3. Definition. Define a bilinear form via

h[M], [N]i := dim HomHn (M,N) for M,N ∈ Hn − ˜ 6.4. Lemma. For φ ∈ Rn (viewed as a character of Bn), set φα := φ(x) for any x ∈ Cα. Then, − − (a) For φ ∈ Rm, ψ ∈ Rn , and γ ∈ OPm+n, X zγ (φ · ψ)γ = φαψβ zαzβ α∈OPm,β∈OPn α∪β=γ

where zα is the order of the centralizer of σ of cycle type α in Sn. (b) X −`(α) −1 hφ, ψi = 2 zα φαψα α∈OP 6.5. Proposition. The character value vanishes unless you are in an even split conjugacy class.

7. The ring Γ (presented by Jinkui Wan)

7.1. Definition. Let x = {x1, x2,...}.

(a) Deﬁne qr = qr(x) via the generating function

X r Y 1 + txi Q(t) = qr(x)t = 1 − txi r≥0 i≥1

10 (b) Let Γ be the Z-subring of the ring of symmetric functions generated by qr, r ≥ 0. (c)Γ Q := Q ⊗Z Γ P r 7.2. Proposition. (a) r+s=n(−1) qrqs = 0 because Q(t)Q(−t) = 1 (b) q = P 2`(α)z−1p where p = P ··· p because ln Q(t) = n α∈OPn α α α α1 α` r P 2pr(x)t r odd r .

7.3. Theorem. (a) ΓQ is a polynomial algebra with polynomial generators p2r−1 for r ≥ 1. (b) {pµ | µ ∈ OP} is a basis for ΓQ

7.4. Definition. Let us deﬁne inner product on ΓQ via −`(α) hpα, pβi := 2 zαδαβ, ∀α, β ∈ OP 7.5. Definition. Deﬁne the (spin) characteristic map to be ch− : R− → Γ Q Q X −1 φ 7→ zα φαpα α∈OPn 7.6. Proposition. (a) ch− is an algebra isomorphism. (b) ch− is an isometry (that is, hφ, ψi = hch−(φ), ch−(ψ)i). Now, we seek to construct the basic spin module.

7.7. Proposition. Hn = C`n o CSn acts on C`n = span{cI | I ⊆ {1, 2, . . . , n}} via ( ci.(ci1 ··· cik ) = cici1 ··· cik

σ.(ci1 ··· cik ) = cσ(i1) ··· cσ(ik) where ci ∈ C`n, σ ∈ Sn, and the action is extended by linearity.

7.8. Proposition. Let σ = σ1 ··· σ` be a cycle decomposition of σ. Then ( ±cI if I is a union of some supports of σ1, . . . , σ` σcI = ±cJ (J 6= I) otherwise

7.9. Example. Let σ = (134)(25) ∈ S5. Then,

σc3c5 = c4c2 but σc1c3c4 = c3c4c1 = c1c3c4 Thus, the character of this action, say ξn, satisﬁes n `(α) ξ (α) = 2 , α ∈ OPn and thus ch−(ξn) = P z−12`(α)p = q . α∈OPn α α n

11 7.10. Definition. For λ ∈ SP, we deﬁne ξλ via the recursive formulas λ X2 ξ(λ1,λ2) = ξλ1 ξλ2 + 2 (−1)iξλ1+iξλ2−i i=1 ( ˆ Pk j (λ1,λj ) (λ2,...,λj ,...,λk) λ j=2(−1) ξ ξ k = `(λ) is even ξ = ˆ Pk j−1 λj (λ1,...,λj ,...,λk) j=1(−1) ξ ξ k = `(λ) is odd

− λ 7.11. Theorem. (a) ch (ξ ) = Qλ, the Schur-Q function (to be deﬁned in the next lecture). n λ − `(λ)−δ(λ) λ o (b) ζ := 2 2 ξ | λ ∈ SPn , where δ(λ) = χ{`(λ) is odd}, is a complete list of simple characters. (c) ζλ is of type M if `(λ) is even and of type Q if `(λ) is odd. (d) The degree of ζλ is   n− `(λ)−δ(λ) n! Y λi − λj 2 2 λ ! ··· λ !  λ + λ  1 ` i

8. Schur-Q functions and related combinatorics (presented by George H. Seelinger) 9. Center of Symmetric Group Algebras and Spin Symmetric Group Algebras (presented by Jinkui Wan)

9.1. Farahat-Higman’s Construction for Sn. Given a permutation σ, we note that its cycle type is not stable under inclusion from Sn ,→ Sn+1.

9.1. Example. Let σ = (134)(2576) ∈ S8. Then, σ has cycle type

(4, 3, 1) =

but, when included into Sn9, σ has cycle type

(4, 3, 1, 1) =

9.2. Definition. Given a cycle σ ∈ Sn, its modiﬁed cycle type, λ, is given by removing the ﬁrst column from its cycle type.

9.3. Example. The modiﬁed cycle type of σ = (134)(2576) ∈ S8 is λ = (3, 2). Note that this is stable with respect to S8 ,→ S9.

12 9.4. Proposition. (a) If σ is of modified type λ, then |λ| is the minimal length for σ as a product of (not necessarily simple) transpositions. (b) If  σ is of modified type λ  τ is of modified type µ , then |ν| ≤ |λ| + |µ| στ is of modified type ν

9.5. Example. (134) = (13)(34) and has modiﬁed type (2).

9.6. Definition. (a) Let Cλ(n) be the conjugacy class of Sn of modiﬁed type λ. Note Cλ(n) = ∅ if n < |λ| + `(λ). (b) Let ( Class sum of Cλ(n) if n ≥ |λ| + `(λ) Cλ(n) := 0 otherwise

9.7. Example. C0(n) = {id} and C(1)(n) contains all transpositions of P Sn. Thus, C(1)(n) = 1≤i

9.8. Proposition. {Cλ(n) | |λ| + `(λ) ≤ n} is a basis for Z(ZSn). 9.9. Definition. Write X ν Cλ(n)Cµ(n) = Aλµ(n)Cν(n) 9.10. Example. 1 C (n)C (n) = 3C (n) + 2C (n) + n(n − 1)C (n) (1) (1) (2) (1,1) 2 0 2 P since C(1)(n) = (ij)(kl) for all transpositions (ij), (kl) in Sn. 9.11. Theorem (Farahat-Higman). Let λ, µ, ν be partitions. Then, ν ν (a) There is a unique polynomial fλµ(x) ∈ Q[x] such that aλµ(n) = ν fλµ(n) for all n ≥ |ν| + `(ν). ν (b) fλµ(x) = 0 unless |ν| ≤ |λ| + |µ| ν ν (c) If |ν| = |λ| + |µ|, then fλµ(x) is a constant. In other words, aλµ(n) is independent of n.

Proof Idea. Let Γ = {(σ, τ) | σ ∈ Cλ(n), τ ∈ Cµ(n), στ ∈ Cν(n)}. Then, once computes ν #Γ aλµ(n) = #Cλ(n) −1 −1 If we let Sn act on Γ by conjugation, that is γ.(σ, τ) = (γσγ , γτγ ), then we get that

Γ = Γ1 ∪ Γ2 ∪ · · · ∪ Γk

13 Without loss of generality, take (σ1, τ1) ∈ Γ1. Then, n! #Γ1 = # of centralizers of (σ1, τ1) in Sn

Suppose γ.(σ1, τ1) = (σ1, τ1). Then, −1 −1 γσ1γ = σ1 and γτ1γ = τ1 Thus,

γ ∈ SSupp(σ1,τ1) × S{1,...,n}\Supp(σ1,τ1) where Supp(σ1, τ1) = {j ∈ {1, . . . , n} | σ1(j) 6= j or τ1(j) 6= j}. So,

# centralizers of (σ1, τ1) in Sn = # centralizers of (σ1, τ1) in SSupp(σ1,τ1)×(n−# Supp(σ1, τ1))! ν Thus, using our formula above for aλµ, we arrive at Finish this for- P mula ν i #Γi aλµ(n) = #Cλ(n) 9.12. Definition. (a) Let B be the ring of polynomials f(x) ∈ Q[x] such that f(n) ∈ Z for all n ∈ Z. (b) Let K be the B-algebra with basis {cλ | λ ∈ P} such that X ν cλcµ := fλµ(x)cν ν∈P We call this ring the Farahat-Higman ring. 9.13. Proposition. We have the following facts (a) K is commutative and associative. (b) K is filtered via deg(cλ) = |λ| for all λ ∈ P. P ν 9.14. Remark. K is not graded because ν∈P fλµ(x)cν is not homogeneous. However, if we say Kr = span{cλ | |λ| ≤ r}, then KrKs ⊆ Kr+s, making K filtered. 9.15. Definition. Let gr K be the associated graded algebra, that is, L gr K is defined by (gr K)r = Kr/Kr−1 and then gr K = r≥0(gr K)r. 9.16. Lemma. We have the following facts. (a) If |λ| + s = m, then ( (m+1)s! (m) Q m (λ)! if `(λ) ≤ s + 1 a = i≥0 i λ,(s) 0 otherwise

where m0(λ) = r + 1 − `(λ). (b) If |λ| + s = |ν|, then

ν X (νi) aλ,(s) = aµ,(s) (i,µ)∈N×P 1≤i≤`(λ) µ∪ν=λ∪(νi)

14 9.17. Proposition. Let λ be a partition and let mi(λ) be the number of mi(λ) i’s in λ such that λ = (i )i≥1. Then, the top degree of cλc(s) is given by

∗ X ν X (ms+|µ|(λ) + 1)(s + |µ| + 1)s! (cλc(s)) = aλ,(s)cν = Q cλ∪(s+|µ|)−µ (s + 1 − `(µ))! mi(µ)! |ν|=|λ|+s µ⊆λ,`(µ)≤s+1 i≥1

m (µ) 9.18. Remark. µ = (i i ) ⊆ λ ⇐⇒ mi(µ) ≤ mi(λ) ∀i ≥ i. 9.19. Example. To illustrate the formula λ ∪ (s + |µ|) − µ, consider cycles σ = (134)(2567)(8) and τ = (28). Then, λ = (3, 2) is the modified cycle type of σ and s = 1 gives the modified cycle type of τ. Then, στ = (134)(28567) which has modified cycle type (4, 2) = (3, 2) ∪ (1 + |(3)|) − (3).

9.20. Corollary. X cλ1 cλ2 ··· cλ` = dλµcµ µDλ and dλλ > 0 in gr K. Thus, c1, c2,... are algebraically independent elements of spanZ{cλ}.

9.21. Proposition. Q ⊗Z gr K is a polynomial algebra generated by c1, c2,... (Note that Z ,→ B as constants.) ∼ 9.22. Remark. (a) There exists a ring isomorphism Λ → gr K send- ∗ ing duals of hλ (images of hλ under a certain automorphism), called gλ, to cλ. See [Mac79, p 132–3]. ∼ ∗ n 2 (b) gr Z(ZSn) = H (Hilb (C ); Z), the cohomology ring of the Hilbert 2 Scheme of points on C , as a Z-algebra. 9.23. Theorem. The homomorphism given by

Πn : K → Z(ZSn) X X fλ(x)cλ 7→ fλ(n)cλ(n) is a surjective homomorphism. P 9.24. Proposition. K is generated by Km := |λ|=m cλ for m ≥ 0.

This tells us that X X Πm(K) = cλ(n) = σ

|λ|=m σ∈Sn # of cycles in σ=n−m and so Z(ZSn) is generated by Πn(K0), Πn(K1),..., Πn(Kn−1).

15 10. Double cover of S˜ n and even split conjugacy classes Recall the short exact sequence

1 → {1, z} → S˜ n → Sn → 1 where  z is central *  2 2 + z = 1, ti = z S˜ n = z, t1, t2, . . . , tn−1 | titi+1ti = ti+1titi+1  titj = ztjti |i − j| > 1 10.1. Definition. Deﬁne the element

xi := titi+1 ··· tn−1tntn−1 ··· ti+1ti which gets mapped to the transposition (i, n) ∈ Sn under the map θn. Then, we let ( z if m = 1 [i1i2 ··· im] := xi1 xim xim−1 ··· xi2 xi1 if m ≥ 2

10.2. Proposition. Every element of S˜ n is of the form q z [i1i2 ··· im][j1j2 ··· jk] ··· | {z } disjoint where q = 0, 1. −1 10.3. Lemma. For λ such that |λ| + `(λ) ≤ n, θn (Cλ(n)) splits if and only if (a) λ has only even parts or (b) λ ∈ SP, |λ| odd, |λ| + `(λ) = n or n − 1.

10.4. Proposition. σ ∈ Sn of modiﬁed type λ is even if and only if |λ| is even. ˜ 10.5. Definition. Let Dλ(n) be the even split conjugacy class in Sn containing [1, 2, . . . , λ1 + 1][λ1 + 2, . . . , λ1 + λ2 + 2] .... −1 10.6. Proposition. (a) θ (Cλ(n)) = Dλ(n) ∪ zDλ(n). (b) {dλ(n) | λ ∈ EP, |λ| + `(λ) ≤ n} is a basis for the even center of − ˜ ZSn = ZSn/hz + 1i. ν 10.7. Definition. Deﬁne bλµ(n) by X ν dλ(n)dµ(n) = bλµ(n)cν(n) ν∈EP 10.8. Example. − d(4)(8)d(2)(8) = 13d(4)(8)−35d(2)(8)−18d(2,2)(8)−7d(6)(8)+2d(4,2)(8) ∈ Z(ZS8 ) 10.9. Theorem (Tysse-Wang). Let λ, µ, ν ∈ EP.

16 ν ν ν (a) There exists a unique gλµ(x) ∈ Q[x] such that bλµ(n) = gλµ(n) for all n ≥ |ν| + `(ν). ν (b) gλµ(x) = 0 unless |ν| ≤ |λ| + |µ| ν (c) If |ν| = |λ| + |µ|, then gλµ(x) is a constnat. 10.10. Definition. Let the spin Farahat-Higman algebra F be a B- algebra with basis {dλ | λ ∈ EP} and X ν dλdµ = gλµ(x)dν ν∈EP which is ﬁltered with respect to deg(dλ) = |λ|.

mi(λ) 10.11. Proposition. Let λ be a partition and rewrite λ = (ii≥1 ). Let s ≥ 0 be event. Then,

X (ms+|µ|(λ) + 1)(s + |µ| + 1)s! (d d )∗ = (−1)`(µ) d λ (s) (s + 1 − `(µ))! Q m (µ)! λ∪(s+|µ|)−µ µ i≥1 i What is this summation 10.12. Corollary. (a) Q ⊗ [ 1 ] gr F is generated by d2, d4,... Z 2 over? (b) There exists an injective homomorphism Q⊗ 1 gr F ,→ Q⊗ K via Z[ 2 ] Z `(λ)c (for λ ∈ EP) dλ 7→ (−1) λ . 10.1. Connections to odd Jucys-Murphy elements. 10.13. Definition. Let us deﬁne k−1 X − Mk := [i, k] ∈ ZSn i=1 10.14. Proposition. We have

(a) MkMl = −MlMk for k 6= l (b) 2 X − Mk = −(k − 1) − [i, j, k] ∈ ZSn 1≤i6=j≤k−1 10.15. Definition. we deﬁne X 2 2 2 − er,n := Mi1 Mi2 ··· Mir ∈ Z(ZSn ) 1≤i1

10.16. Proposition. (a) er,n has top degree 2r P (b) er,n = λ∈EP,|λ|+`(λ)≤n Aλ(n)dλ(n) for some Aλ(n). (c) Aλ(n) is the coeﬃcient of [1, 2, . . . , λ1 +1][λ1 +2, . . . , λ1 +λ2 +2] ··· in er,n and is independent of n.

10.17. Definition. In light of the proposition above, we write Aλ := Aλ(n) and deﬁne ∗ X er := Aλdλ ∈ F λ∈EP,|λ|=2r

17 ∗ ∗ 10.18. Example. e1 = −d2 and e2 = d(2,2) − 2d4. Q `(λ) i≥1 c λi 10.19. Proposition. Aλ = (−1) 2 where c0 = 1 and cr = 1 2r r+1 r are the Cartan numbers. 1 ∗ ∗ ∗ 10.20. Theorem. B 2 ⊗B F is generated by e1, e2, e3,... 10.21. Corollary. Via the surjective homomorphism 1 1 ⊗ → Z( S−) B 2 B F Z 2 n X X fλ(x)dλ 7→ fλ(n)dλ(n) λ∈EP λ∈EP 1 − the even center of Z 2 Sn is generated by (the top degree of) er,n.

11. Schur-Sergeev duality for q(n) (presented by Chris Chung)

12. Seminormal form construction for irreducible Hn-modules (presented by Jinkui Wan) A review for the symmetric group case was presented, but not written up here yet.

12.1. Definition. We deﬁne the Jucys-Murphy elements in Hn = C`n o CSn as X Jk := (1 + cjck)(jk), 1≤j

(a) JkJl = JlJk for l ≤ k 6= l ≤ n. (b) ckJk = −Jkck and clJk = Jkcl for k 6= l. (c) skJk = Jk+1sk − (1 + ckck+1) for 1 ≤ k ≤ n − 1 and slJk = Jksl for k 6= l, l + 1. 12.3. Definition. The degenerate aﬃne Hecke-Cliﬀord algebra is given by

Hˆn = hs1, . . . , sn−1, c1, . . . , cn, x1, . . . , xni with additional relations  xkxl = xlxk  s x = x s − (1 + c c )  k k k+1 k k k+1 sixk = xksi k 6= i, i + 1  xkck = −ckxk  xkcl = clxk l 6= k

18 ˆ 12.4. Proposition. There exists a projection π : Hn Hn such that

π(si) = si, π(ck) = ck, π(xl) = Jl

In particular, π(x1) = J1 = 0, so x1 ∈ ker π.

In fact, ker π = hx1i and so Hn-Mod can be identiﬁed as the subcategory of Hˆn-Mod on which x1 = 0. 12.5. Theorem (PBW Theorem). We have that

α1 αn β1 βn {x1 ··· xn c1 ··· cn w | αi ∈ Z+, βi ∈ {0, 1}, 1 ≤ i ≤ n, w ∈ Sn} is a basis for Hˆn.

12.6. Corollary (Corollary of PBW Theorem). The subalgebra of Hˆn generated by x1, . . . , xn, c1, . . . , cn is isomorphic to

c c c C`n ⊗C[x1, . . . , xn]/hxkck = −ckxk, xkcl = clxk, l 6= ki = P1 ⊗ P1 ⊗ · · · ⊗ P1 | {z } n copies

c where P1 = hx1, c1i.

2 2 2 Sn ˆ 12.7. Proposition. The Sn fixed points C[x1, x2, . . . , xn] ⊆ Center of Hn. 2 2 12.8. Proposition. (a) The eigenvalues of x1, . . . , xn are of the form q(i) := i(i + 1) for i ∈ Z+. 2 ˆ (b) If all the eigenvalues of xj on a finite dimensional Hn-module M for a fixed j are of the form q(i), then M is integral.

The second part of the proposition follows from the intertwining elements.

12.9. Definition. Let intertwining element Φk ∈ Hˆn be given by

2 2 Φk := sk(xk − xk+1) + (xk + xk+1) + ckck+1(xk − xk+1) 2 2 = (xk+1 − xk)sk − (xk + xk+1) − ckck+1(xk − xk+1)

Note, the second equality follows from the fact that

2 2 skxk = xk+1skxk − (1 + ckck+1)xk = xk+1sk − (1 + ckck+1)(1 + xk) and, using conjugation by sk,

2 2 2 2 xksk = skxk+1−sk(1+ckck+1)(1+xk)sk =⇒ −skxk+1 = −xksk−sk(1+ckck+1)(1+xk)sk Therefore, we get

2 2 2 2 sk(xk −xk+1) = (xk+1 −xk)sk −(1+ckck+1)(1+xk)−sk(1+ckck+1)(1+xk)sk

19 12.10. Proposition. We have the following useful relations for the intertwining elements.  Φ Φ = Φ Φ |k − l| > 1  k l l k  ΦkΦk+1Φk = Φk+1ΦkΦk+1  2 2 2 2 2 2 Φk = 2(xk + xk+1) − (xk − xk+1)  Φkxk = xk+1Φk, Φkxk+1 = xkΦk, Φkxl = xlΦk l 6= k, k + 1  Φkck = ck+1Φk, Φkck+1 = ckΦk, Φkcl = clΦk l 6= k, k + 1

2 2 12.11. Proposition. If v is some eigenvector of xj+1, that is, if xj+1v = 2 2 av, then xj Φjv = aΦjv, so Φjv is an xj eigenvector with the same eigen- value.

2 2 Proof. Since xj Φj = Φjxj+1, we get that 2 2 xj Φjx = Φjxj+1v = Φjav = aΦjx

12.12. Definition. A ﬁnite dimensional Hˆn-module M is called com- pletely splittable (CS) if x1, x2, . . . , xn act semisimply, ie the actions of x1, . . . , xn can be diagonalized simultaneously.

12.13. Proposition. Every integral CS Hˆn-module M can be decom- posed as M M = Mi n i∈Z+ where 2 Mi = {v ∈ M | xkv = q(ik)v, 1 ≤ k ≤ n} 2 2 2 is the common eigenspace of x1, x2, . . . , xn with eigenvalues q(i1), q(i2), . . . , q(ik). Furthermore, deﬁne n wt(M) := {i ∈ Z+ | Mi 6= 0}

Our goal is to describe wt(M) for integral irreducible CS Hˆn-modules. n 12.14. Lemma. If i = (i1, . . . , in) ∈ Z+ is in wt(M) for some integral irreducible CS Hˆn-module, then ik 6= ik+1 for all 1 ≤ k ≤ n − 1.

2 2 Proof. Suppose ik = ik+1 so that xkv = xk+1v for v ∈ Mi. Since 4 2 2 2 2 2 2 xksk−2q(ik)xksk+q(ik) sk = xk(skxk+1−sk(1+ckck+1)(1+xk)sk)−2q(ik)xksk+q(ik) sk

Somehow we get that skv ∈ Mi. Then, we get ﬁgure this out 2 =⇒ xkskv = q(ik)skv 2 =⇒ (xk − q(ik))skv = 0 2 =⇒ (skxk+1 − xk(1 − ckck+1) − (1 − ckck+1)xk+1)v − q(ik)skv = 0

20 2 However, xk+1v = q(ik+1)v = q(ik)v by assumption, so

=⇒ (xk(1 − ckck+1) + (1 − ckck+1)xk)v = 0 2 2 =⇒ 2(xk + xk+1)v = 0 =⇒ ik = ik+1 = 0 2 2 =⇒ xkv = 0 = xk+1v =⇒ xkv = 0 = xk+1v since M is CS =⇒ v = 0 since Finish this n proof. 12.15. Lemma. Suppose i = (i1, . . . , in) ∈ wt(M) ⊆ Z+ for some integral, irreducible, CS Hˆn-module. Fix 1 ≤ k ≤ n − 1.

(a) If ik 6= ik+1 ± 1, then Φkz 6= 0 for all 0 6= z ∈ Mi. (b) If ik = ik+1 ± 1, then Φk = 0 on Mi. 2 2 2 2 2 2 Proof. Since Φk = 2(xk + xk+1) − (xk − xk+1) , then 2 2 2 Φkz = (2(q(ik) + q(ik+1)) − q(ik) + 2q(ik)q(ik+1) − q(ik+1) )z = 0 if and only if 2 (q(ik) − q(ik+1)) = 2(q(ik) + q(ik+1)) Now, if we write ik = ik+1 + c, then ( 2 2 ( 2 q(ik) = ik+1 + 2cik+1 + c + ik+1 + c q(ik) − q(ik+1) = 2cik+1 + c + c 2 =⇒ 2 2 q(ik+1) = ik+1 + ik+1 q(ik) + q(ik+1) = 2ik+1 + 2cik+1 + c + 2ik+1 + c From here, one checks that c = ±1 certainly gives solutions independent of 2 ik+1. Thus, ik = ik+1 ± 1 =⇒ Φkz = 0. Furthermore, there are other formal solutions to these equations, namely c = −2ik+1 and c = −2(ik+1 + 1), but since ik+1 ≥ 0, this would force ik < 0 unless ik = ik+1 = 0, which is not admissible by the previous lemma. 2 So, to prove the second part, since Φkz = 0, it must be that if Φkz 6= 0 and so Φkz ∈ Mski . Then, there exists a minimal sequence Φj1 ,..., Φjr such Why is this that Φj1 ··· Φjr Φkz ∈ Mi since M is irreducible. Then, if σ = sj1 ··· sjr sk ∈ true? Sn, it must be that σ · i = i. If one assumes σ 6= 1, this leads to a violation of Lemma 12.14 with some work. Then, using the exchange condition for 2 Coxeter groups, one shows that r = 1 which gives j1 = k, so Φkz 6= 0, contradicting what we showed above.

12.16. Remark. Suppose V is an integral, irreducible Hˆn-module. Let ˆ H(n−r,1k) = hs1, . . . , sn−r−1, c1, . . . , cn, x1, . . . , xri. Then,

V is CS ⇐⇒ ∀i ∈ wt(V ), 1 ≤ k ≤ n − 1, ik 6= ik+1

Hˆn ⇐⇒ Res ˆ V is semisimple ∀1 ≤ r ≤ n H(n−r,1r)

Hˆn ⇐⇒ Res ˆ V is semisimple H(1k,n−k−r,1r)

21 The second Hˆ k r 12.17. Corollary. Suppose i ∈ wt(V ) for some integral, irreducible, (1 ,n−k−r,1 ) is not deﬁned. CS Hˆn-module V . If ik = ik+2 for some 1 ≤ k ≤ n − 2, then ik = ik+2 = 0 and ik+1 = 1.

Proof. If ik 6= ik+1 ± 1, then sk · i ∈ wt V by the ﬁrst part of 12.15, but sk · i = (··· , ik+1, ik, ik+2 ··· ) and, by assumption, ik = ik+2 contradicting 12.14. So, it must be that ik+1 = ik ± 1 and so Φk = 0 = Φk+1 on Vi by the second part of 12.15. Thus, for z ∈ V , ( Φkz = 0 =⇒ (q(ik) − q(ik+1))skz = −((xk + xk+1) + ckck+1(xk − xk+1))z Φk+1z = 0 =⇒ (q(ik+1) − q(ik+2))sk+1z = −((xk+1 + xk+2) + ck+1ck+2(xk+1 − xk+2))z which gives us the sk and sk+1 actions on Vi. From here, we can use the braid relation sksk+1sk = sk+1sksk+1 to arrive at the equality 2 2 ((xk + xk+2)(6xk+1 + 2xkxk+2) + ckck+2(xk − xk+2)(6xk+1 − 2xkxk+2))z = 0 p Now, xkz = ± q(ik)z. Furthermore, since ik = ik+2, we have that either xkz = xk+2z or xkz = −xk+2z. Decompose

Vi = W1 ⊕ W2 where W1 := {z ∈ Vi | xkz = xk+2z} and W2 := {z ∈ Vi | xkz = −xk+2z}. Now, we can break up our braid relation equality. 2 2 z ∈ W1 =⇒ 0 = ((xk + xk+2)(6xk+1 + 2xkxk+2) + ckck+2(xk − xk+2)(6xk+1 − 2xkxk+2))z 2 2 = 2xk(6xk+1 + 2xk)z p = 2 q(ik)(6q(ik+1) + 2q(ik))z 2 2 z ∈ W2 =⇒ 0 = ((xk + xk+2)(6xk+1 + 2xkxk+2) + ckck+2(xk − xk+2)(6xk+1 − 2xkxk+2))z 2 2 = ckck+22xk(6xk+1 + 2xk)z p = ckck+2(2 q(ik)(6q(ik+1) + 2q(ik)))z Thus, we obtain that p 2 q(ik)(6q(ik+1) + 2q(ik)) = 0

Moreover, we know that ik+1 = ik ± 1, so we get (p ik(ik + 1)(6(ik − 1)ik + 2(ik + 1)ik) = 0 if ik+1 = ik − 1 p ik(ik + 1)(6(ik + 1)(ik + 2) + 2(ik + 1)ik) = 0 if ik+1 = ik + 1

There are no solutions to the ﬁrst equation that give ik and ik+1 as non- negative integers and the only such solution for the second equation is ik = 0 and ik+1 = 1. In conclusion, we have the following. 12.18. Theorem. Let W(n) be the set of weights of all integral irreducible CS Hˆn-modules and let i ∈ W(n). Then,

22 (a) ik 6= ik+1 for all 1 ≤ k ≤ n − 1. (b) If ik = i` = 0 for some 1 ≤ k < ` ≤ n, then 1 ∈ {ik+1, . . . , i`−1}. (c) If ik = i` ≥ 1 for some 1 ≤ k < ` ≤ n, then {ik − 1, ik + 1} ⊆ {ik+1, . . . , i`−1}. Proof. The ﬁrst part is just a restatement of 12.14. For the next part, assume 1 6∈ {ik+1, . . . , i`−1}. Then, we can swap indices using ?? to get new weights until we obtain a weight of the form (..., 0, 0,...) which is not allowed by the previous part.

For the last part, let u = ik = i` ≥ 1 be such that k − ` is minimal among such occurrences. If only one of u + 1, u − 1 appear in between ik and i`, then it must appear twice because, otherwise, we could swap indices using 12.15 to get new weights until we are of the form (. . . , u, u ± 1, u, . . .) which is not a weight by 12.17 since u ≥ 1. Thus, we have violated the minimality of our choice of u. Now, we wish to describe a bijection between W(n) and standard skew shifted tableaux of size n. 12.19. Definition. Given strict partitions ν ⊆ ξ, the skew shifted Fer- rers diagram ξ/ν is given by removing the boxes of ν from ξ. 12.20. Example. Consider (3, 2) ⊆ (5, 4, 2, 1). Then,

7→

We now illustrate the bijection by example. 12.21. Example. Let i = (1, 0, 3, 2, 1, 0, 6). Then, we construct our standard skew shifted tableau in steps by adding a box of content ik labelled k on the kth step. 1 (1) 7→ on content 1 diagonal 1 (1, 0) 7→ contents 1, 0 2 3 (1, 0, 3) 7→ 1 2

23 3 (1, 0, 3, 2) 7→ 1 4 2 . . 3 (1, 0, 3, 2, 1, 0) 7→ 1 4 2 5 6

(1, 0, 3, 2, 1, 0, 6) 7→ 7 3 1 4 2 5 6

So, our ﬁnal answer has outer shape ξ = (7, 4, 3, 2, 1) and inner shape ν = (6, 3, 1): 7 3 1 4 2 5 6

12.22. Definition. Given i ∈ wt V , let T (i) be the corresponding standard skew shifted tableau. 12.23. Proposition. If i, j ∈ wt(V ) for some integral irreducible CS Hˆn-module V , then T (i) and T (j) have the same shape.

Proof. If (i1, . . . , ik, ik+1, . . . , in), then (i1, . . . , ik−1, ik+1, ik, . . . , in) is a weight only if ik 6= ik+1 ± 1 by ??. However, under such a condition, it What is the ref- does not matter in which order we add the boxes corresponding to ik and erence here? ik+1. Thus, these two weights will yield the same shape. 12.24. Definition. Given ξ/ν a skew-shifted Ferrer’s diagram of size n, we deﬁne F(ξ/ν) := {T | T a standard Young tableau of shape ξ/ν} and ξ/ν M Dˆ := C`nvT T∈F(ξ/ν)

24 as a vector space with actions p xkvT = q(c(Tk))vT where c(Tk) is the content of the box labelled by k in T and where C`n acts by multiplication on the left. 12.25. Proposition. We have ! s 1 1 2(q(c(T ))) + q(c(T )) s v = + c c v + 1 − k+1 k v k T p p p p k k+1 T 2 skT q(c(Tk+1)) − q(c(Tk)) q(c(Tk+1)) + q(c(Tk)) (q(c(Tk+1)) − q(c(Tk)))

Proof. This fact follows formally from the fact that ΦkvT = avskT for some scalar a if skT is standard. Actually do this ξ/ν proof. 12.26. Corollary. Dˆ is an integral CS Hˆn-module.

Proof. Mainly, one needs to check the Coxeter relations. Fill in this proof. 13. Spin Kostka Polynomials

13.1. Definition. Let ν ∈ SP and µ be a partition. Then, the spin Kostka polynomials are the transition polynomials from the Hall-Littlewood P -functions to the Q-Schur functions. In other words, X − Qν(x) = Kνµ(t)Pµ(x; t) µ

If we also let bνλ be such that X Qν(x) = bνλsλ(x) λ then we get 13.2. Proposition. For ν ∈ SP and µ a partition, − X Kνµ(t) = bνλKλµ(t) λ where Kλµ(t) are the Kostka-Foulkes polynomials. Proof. By deﬁnition of Kostka-Foulkes, we have X sλ = Kλµ(t)Pµ(x; t) µ and so X X X Kνµ(t)Pµ(x; t) = Qν(x) = bνλsλ = bνλKλµ(t)Pµ(x; t) µ λ λ,µ

25 Bibliography

[CW12] S.-J. Cheng and W. Wang, Dualities and Representations of Lie Superalgebras, 2012. [Kle05] A. Kleshchev, Linear and Projective Representations of Symmetric Groups, 2005. [Mac79] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 1979. 2nd Edition, 1995. [Sag87] B. E. Sagan, Shifted Tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Combin. Theory Ser. A (1987), 62–103. [WW12] J. Wan and W. Wang, Lectures on Spin Representation Theory of Symmetric Groups, Bull. Inst. Math. Acad. Sin. (2012), 91–164.