The Terwilliger algebra of the twisted Grassmann graph: the thin case

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this"thin"case.


Introduction
In 2005, Van Dam and Koolen [5] discovered a new infinite family of distance-regular graphs with unbounded diameter, which they call the twisted Grassmann graphs. Let q be a prime power, and let D be an integer at least two. Fix a hyperplane H of the vector space F 2D+1 q over the finite field F q . Let X ′ be the set of (D + 1)-dimensional subspaces of F 2D+1 q not contained in H, and let X ′′ be the set of (D − 1)-dimensional subspaces of H. The twisted Grassmann graphJ q (2D + 1, D) has vertex set X = X ′ ⊔ X ′′ , where two vertices y and z are adjacent whenever dim y + dim z − 2 dim y ∩ z = 2. (1) The graphJ q (2D +1, D) has the same intersection array as the Grassmann graph J q (2D + 1, D) on the set of D-dimensional subspaces of F 2D+1 q . A particularly interesting feature of We then show that, for every irreducibleH-moduleW, the subspaceẼ * W is either an irreducible T (x)-module, or the direct sum of two irreducible T (x)-modules. We also find the isomorphisms among these irreducible T (x)-modules. In view of the semisimplicity of T (x), this completes the classification of the irreducible T (x)-modules. Our main results are Theorems 4.2,4.3,and 4.4. See also the comments after Theorem 4.4. Throughout this paper, we fix a prime power q and use the following notation: (1 − αq ℓ ) (α ∈ C, i ∈ Z 0 ), m n = m n q = (q) m (q) n (q) m−n (m, n ∈ Z 0 , m n).
For every non-empty finite set S, we let Mat S (C) denote the C-algebra of complex matrices with rows and columns indexed by S, and we also let CS denote the C-vector space with basis S, on which Mat S (C) acts from the left in the standard manner.

The algebra H
Let a and b be non-negative integers, and let P be the set of all subspaces of F a+b q . We will always fix x ∈ P with dim x = a. For 0 i a and 0 j b, let P i,j = {y ∈ P : dim x ∩ y = i, dim y = i + j}.
We note that the P i,j give a partition of P , and that (cf. [3,Lemma 9
In particular, if we let s q (n) denote the number of subspaces of F n q , then we have For convenience, we set P i,j := ∅ for i, j ∈ Z unless 0 i a and 0 j b. For 0 i a and 0 j b, let E * i,j ∈ Mat P (C) be the diagonal matrix with (y, y)-entry (E * i,j ) y,y = 1 if y ∈ P i,j , 0 otherwise, (y ∈ P ).
We note that (L 1 ) T = R 1 and (L 2 ) T = R 2 , where T denotes transpose. Let H be the subalgebra of Mat P (C) generated by L 1 , L 2 , R 1 , R 2 , and all the E * i,j . The algebra H is semisimple as it is closed under conjugate-transpose. We note that every irreducible H-module appears in CP up to isomorphism.
Let W be an irreducible H-module. Let We call (ν, µ) (resp. (ν ′ , µ ′ )) the lower endpoint (resp. upper endpoint) of W. It is known and call ρ the index of W.
Let W be an irreducible H-module with lower endpoint (ν, µ) and index ρ. Then W has a basis w i,j (ν i a − ν − ρ, µ j b − µ + ρ) such that w i,j ∈ E * i,j W, and for all i, j, where we set w ν−1,j = w a−ν−ρ+1,j = w i,µ−1 = w i,b−µ+ρ+1 := 0. In particular, the isomorphism class of W is determined by ν, µ, and ρ. Moreover, the multiplicity m ν,µ,ρ of W in CP is given by Remark 2.2. The generators of the algebra H commute with the action on P of the maximal parabolic subgroup of GL(F a+b q ) stabilizing the subspace x. Hence H is a subalgebra of the centralizer algebra of this parabolic subgroup. However, by comparing Theorem 2.1 with the results of Dunkl [7] we can show that the two algebras are in fact equal. In particular, the formula for the multiplicity m ν,µ,ρ above agrees with [7,Proposition 4.15] under the replacement (ν, µ, ρ) → (m, n + r, r). See also [26].

The algebraH
Let a and b be as in the previous section. LetP be the set of all subspaces of F a+b+1 q . We will again fix x ∈P with dim x = a, and we will also fix a hyperplane H ∈P containing x. For 0 i a, 0 j b, and 0 k 1, let Then theP i,j,k give a partition ofP , and For convenience, we setP i,j,k := ∅ for i, j, k ∈ Z unless 0 i a, 0 j b, and 0 k 1.
For 0 i a, 0 j b, and 0 k 1, letẼ * i,j,k ∈ MatP (C) be the diagonal matrix with (y, y)-entry We note that (L 1 ) T =R 1 , (L 2 ) T =R 2 , and (L 3 ) T =R 3 . LetH be the subalgebra of MatP (C) generated byL 1 ,L 2 ,L 3 ,R 1 ,R 2 ,R 3 , and all theẼ * i,j,k . The algebraH is semisimple as it is closed under conjugate-transpose. We note that every irreduciblẽ H-module appears in CP up to isomorphism.
Our goal in this section is to describe the irreducibleH-modules. To this end, we extend some of the results of Srinivasan [15]. Let G be the subgroup of SL(F a+b+1 q ) consisting of the elements which fix every vector in H. If we fix a basis u 1 , . . . , u a+b+1 of F a+b+1 q such that H = span{u 1 , . . . , u a+b }, then the matrices representing the elements of G with respect to this basis are of the form  Observe that G is abelian and is isomorphic to the additive group F a+b q , and that theP i,j,k are G-invariant. LetP Thus,P 0 is the set of subspaces of H, andP 1 is the set of subspaces of F a+b+1 q not contained in H. We define the equivalence relation ∼ onP 1 by We observe that the equivalence classes of this relation are precisely the G-orbits onP 1 . For every y ∈P 1 , let G y denote the stabilizer of y in G.
Let G be the character group of G with trivial character 1 G . For 0 i a and 0 j b, let ψ i,j be the permutation character of G onP i,j,1 . Note that (g − id)(F a+b+1 q ) is a one-dimensional subspace of H for every g ∈ G with g = id (cf. (3)). The elements of G such that (g − id)(F a+b+1 q ) ⊂ x form a subgroup of G of order q a , which we denote by K. The following extends [15,Theorem 2.3], and the proof is straightforward. (i) For 0 i a, 0 j b, and g ∈ G, we have (ii) For 0 i a, 0 j b, and χ ∈ G, we have where [·, ·] denotes the usual inner product of characters.
For 0 i a, 0 j b, and χ ∈ G, let M(χ) and M(χ) i,j be the homogeneous components of χ in CP 1 and CP i,j,1 , respectively. Note that Hence it follows from Lemma 3.1 (ii) that by virtue of (2). Moreover, Observe that G acts trivially onP 0 , and that the generators ofH commute with the action of G, from which it follows that CP 0 M(1 G ) and the M(χ) (χ = 1 G ) areH-modules. For χ ∈ G, let Note that the e χ are the (central) primitive idempotents of the group algebra CG. In particular, we have For the rest of this section, we will fix u ∈P 0,0,1 . For y ∈P 0 , we will use y ∨ u to denote the subspace of F a+b+1 q spanned by y and u, in order to avoid confusion with the addition in CP . The following is essentially from [15, Theorem 2.5 (i)-(iii)].  (iii): Note that the subspaces y ∨ u (y ∈P i,j,0 ) form a complete set of representatives of the equivalence classes inP i,j,1 . From (i) above and (6), it follows that these vectors span M(χ) i,j . Moreover, they have mutually disjoint supports, and hence are linearly independent.
For any subspaces with respect to the fixed subspace x ′ as in the previous section. (For these matrices, the underlying subspaces x ′ and H ′ will be clear from the context.) We will also consider the corresponding algebra H(x ′ , H ′ ) generated by these matrices. The following extends [15, Proposition 3.4. The following hold: where we have, on CP (x, H), H) and M(1 G ) have the same dimension by (5), it follows thatR 3 gives a vector space isomorphism from CP (x, H) to M(1 G ). The other identities are easily verified.
Let y ∈P i,j,0 be such that y ⊂ H χ . On the one hand, we have On the other hand, sinceR 1 and e χ commute, we havẽ However, in the last sum above, we have Θ χ z = 0 unless z ⊂ H χ . Hence it follows that the above two vectors are equal. This proves thatR and alsoL In the last sum above, for each z, there are exactly q choices for w. Pick any such w. Then since w ∼ z ∨ u, there exists g ∈ G such that w = g(z ∨ u). However, since both w and z ∨ u are subspaces of y ∨ u, this g must fix y ∨ u, i.e., g ∈ G y∨u . On the other hand, recall that Θ χ y = 0 since y ⊂ H χ . Hence it follows from Lemma 3.2 that χ| Gy∨u = 1 Gy∨u .
(iii): Similar to the proof of (ii) above.
LetW be an irreducibleH-module with lower endpoint (ν, µ, τ ) and index ρ. Then the isomorphism class ofW is determined by ν, µ, τ , and ρ, and the following hold: such that w i,j,k ∈Ẽ * i,j,kW , and for all i, j, k, where we set w i,j,k := 0 if (i, j, k) is outside the above range. Moreover, in this case, the multiplicity m ν,µ,0,ρ ofW in CP is given by for all i, j, where we set w i,j,1 := 0 if (i, j) is outside the above range. Moreover, in this case, the multiplicity m ν,µ,1,ρ ofW in CP is given by Proof. First, recall that CP 0 M(1 G ) is anH-module. Recall also the algebra H(x, H) acting on CP (x, H) = CP 0 . Let W be an irreducible H(x, H)-module in CP (x, H) with lower endpoint (ν, µ) and index ρ, and let the basis vectors w i,j be as in Theorem 2.1. For ν i a − ν − ρ and µ j b − µ + ρ, let w i,j,0 = w i,j and w i,j,1 =R 3 w i,j . Then from Proposition 3.4 (i) it follows that the w i,j,k are non-zero and are linearly independent, and that they form a basis of anH-module, which we denote byW. It is also immediate to see that the actions of the generators on the w i,j,k are as in (i). We now claim thatW is irreducible. SinceH is semisimple,W is a direct sum of irreducibleH-submodules. SinceẼ * ν,µ,0W = span{w ν,µ,0 } = 0, there is an irreducibleH-submoduleŨ ofW such that E * ν,µ,0Ũ = 0. Then we have w ν,µ,0 ∈Ũ, and henceW =Hw ν,µ,0 ⊂Ũ , i.e.,W =Ũ. It follows thatW is irreducible. We note thatW has lower endpoint (ν, µ, 0) and index ρ.
Third, let χ ∈ G with χ| K = 1 K . In this case, we consider the algebra H(x ∩ H χ , H χ ), where H χ ∈P a−1,b,0 is from Proposition 3.4 (iii). We again argue as above, but we start from an irreducible H(x ∩ H χ , H χ )-module W with lower endpoint (ν, µ) and index ρ − 1. Then the actions of the generators on the w i,j,1 are again as in (ii), and the irreduciblẽ H-moduleW has lower endpoint (ν, µ, 1) and index ρ.
The formulas for the multiplicities follow from Theorem 2.1. Note that we obtain isomorphic irreducibleH-modules from the second and the third cases above, so that the multiplicity in (ii) is computed by adding two terms, each multiplied by the number of respective characters, and then simplifying. The other statements are also verified using Theorem 2.1.

The Terwilliger algebra T = T (x)
We now turn to the discussions on the Terwilliger algebra T = T (x) of the twisted Grassmann graphJ q (2D + 1, D), where we choose the base vertex x from X ′′ . We mentioned in Introduction thatJ q (2D + 1, D) has the same intersection array as the Grassmann graph J q (2D + 1, D). In particular, it is an example of a Q-polynomial distance-regular graph with diameter D. (For the background information on distance-regular graphs, we refer to [2,3,6,9].) The eigenvalues of J q (2D + 1, D), and hence ofJ q (2D + 1, D), are given in [3, Theorem 9.3.3] as follows:

Recall the diagonal matrices E
. We note that two vertices y and z are at distance i if and only if (cf. (1)) dim y + dim z − 2 dim y ∩ z = 2i.
For 0 i D, let E i ∈ Mat X (C) be the orthogonal projection onto the eigenspace of the adjacency matrix A with eigenvalue θ i . Let W be an irreducible T -module. Define the support and the dual support of W by Here, (8) and (9) Observe that the first two inequalities in (8) are consequences of (9) and (10), so that we may replace (8) in the definition of Ω by the following: It will also be important to consider the following (d + 1) × (d + 1) matrix having four free parameters (besides q and d): where and where we assume that h, r, s = 0, that rq i , sq i /r = 1 (1 i d), and that sq i = 1 (2 i 2d). We note that b i−1 c i = 0 (1 i d). This matrix is the standardized form of one of the operators of a Leonard system of dual q-Hahn type; cf. [25,Example 5.5].
With the notation of [20,Section 2], this also corresponds to Case (I) with r 1 = s * = 0. See also [24,Theorem 17.7]. The eigenvalues of the matrix (12) are given by In particular, if we fix h, s, and λ 0 , then the matrices (12)  if and only if r = r ′ , in which case S is a non-zero scalar matrix.
Proof. Observe that the a i are linear in r provided that d > 0. Since conjugation by S does not change the diagonal entries, the result follows.
If d = 0 then the matrix (12) is in fact independent of h, r, and s, but we will still include these for convenience of the descriptions. For the rest of this paper, we retain the notation of Section 3 with where we take the base vertex x ∈ X ′′ as the fixed subspace x ∈P in Section 3, and similarly for the hyperplane H. Then we have Note thatẼ * is the orthogonal projection onto CX ⊂ CP . We also identify Mat X (C) withẼ * MatP (C)Ẽ * in the obvious manner. With this notation, the adjacency matrix A ∈ Mat X (C) ofJ q (2D + 1, D) is written as A =Ẽ * ′ AẼ * ′ +Ẽ * ′ AẼ * ′′ +Ẽ * ′′ AẼ * ′ +Ẽ * ′′ AẼ * ′′ , and direct computations show that Moreover, we have (cf. (7)) whereẼ * D,0,1 =Ẽ * −1,D,0 := 0. It follows that T is a subalgebra ofẼ * HẼ * . We will now find all the irreducible T -modules in CX. First, letW be an irreduciblẽ H-module in CP with lower endpoint (ν, µ, 0) and index ρ, where we recall from Theorem 3.5 that ν, µ, and ρ satisfy In particular, ν + µ D. Let the basis vectors w i,j,k ofW be as in Theorem 3.5 (i). Note thatẼ * W =Ẽ * ′W Ẽ * ′′W , and that E * ′W = span w D−i,i,1 : max{ν + ρ + 1, µ} i min{D − ν, D − µ + ρ + 1} , where we always haveẼ * ′W = 0, whereasẼ * ′′W = 0 precisely when ν + µ < D. Let for max{ν + ρ, µ} i min{D − ν, D − µ + ρ + 1}, and let where we set w i,j,k := 0 whenever (i, j, k) is outside the parameter range. Observe by (18) that w i , w i ∈ E * iW . We define the subspaces W 1 and W 2 ofẼ * W by W 1 = span w i : max{ν + ρ, µ} i min{D − ν, D − µ + ρ + 1} , where we always have W 1 = 0, whereas W 2 = 0 precisely when ρ < D − 2ν − 1. Then we haveẼ Moreover, it follows from Theorem 3.5 (i) and (14)- (17) that and for all i, where we understand that w i = w i = 0 whenever they are undefined. It follows that W 1 and W 2 are T -modules. We now claim that W 1 and W 2 are thin irreducible T -modules. (For W 2 , the claim holds under the additional assumption that ρ < D − 2ν − 1; otherwise we have W 2 = 0.) Let U be a non-zero T -submodule of W 1 . Since U is closed under the E * i , and since E * i W 1 = span{w i }, it follows that U is spanned by some of the w i . Suppose that w i ∈ U. Observe that the coefficients of w i±1 in Aw i are non-zero whenever w i±1 are defined. In other words, E * i±1 Aw i are non-zero scalar multiples of w i±1 , and hence w i±1 ∈ U. By repeating this argument, it follows that U contains all the basis vectors of W 1 , i.e., U = W 1 . Hence W 1 is an irreducible T -module, and it is clear that W 1 is thin. The same proof works for W 2 as well.
The endpoint ǫ and the diameter d of W 1 are given by Consider the matrix (12) with parameters Let γ i+1 denote the coefficient of w i+1 in Aw i ; cf. (21). We mentioned above that γ i+1 = 0 Then we can routinely verify that A(q, d; h, r, s, λ 0 ) with the above parameters gives the matrix representing A| W 1 with respect to the v i , and that (cf. (13)) It follows that the dual endpoint ǫ * of W 1 is given by where V 1,0 and V 1,1 are T -submodules of V 1 such that the following hold: (i) For the irreducible T -submodules W in V 1,0 , the endpoint ǫ, dual endpoint ǫ * , and the diameter d range over the set (cf. (11)) Every W is thin and has a basis v i (0 i d) such that v i ∈ E * ǫ+i W for all i, and that the matrix representing A| W with respect to it agrees with the matrix (12) with parameters The isomorphism classes in V 1,0 are determined by ǫ, ǫ * , and d, and the corresponding multiplicity m 1,0 ǫ,ǫ * ,d in V 1,0 is given by (ii) Similar statements to (i) above hold for V 1,1 with r = q ǫ * −D−1 , where we replace Ω 1,0 and m 1,0 ǫ,ǫ * ,d by Ω 1,1 and m 1,1 ǫ,ǫ * ,d , respectively, where Proof. We let V 1,0 (resp. V 1,1 ) be the sum of the W 1 for which the correspondingW satisfy ν + ρ µ (resp. ν + ρ < µ). All the computations are routinely done using (19), (20), and Theorem 3.5 (i).
We have and hence Theorems 4.2, 4.3, and 4.4 give all the irreducibleH-modules up to isomorphism. In view of Lemma 4.1, it follows that • Irreducible T -modules in V 1,0 V 2,0 V 3,0 are isomorphic if and only if they have the same ǫ, ǫ * , and d.
• Irreducible T -modules in V 1,1 V 2,1 V 3,1 are isomorphic if and only if they have the same ǫ, ǫ * , and d.
• Irreducible T -modules with d = 0 are isomorphic if and only if they have the same ǫ and ǫ * .
• There are no other isomorphisms.
By these comments, we may compute the multiplicity of an irreducible T -module in CX simply by summing up the corresponding multiplicities in the above summands, but we omit the formulas as these seem too complicated.