Quasi-uniform Posets and Leonard Pairs Associated With Singular Linear Spaces Over Finite Fields

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1 46 1ff ffi ± μ Vol. 46, No Ω1 ADVANCES IN MATHEMATICS (CHINA) Jan., 2017 doi: /sxjz b Quasi-uniform Posets and Leonard Pairs Associated With Singular Linear Spaces Over Finite Fields LI Zengti, ZHANG Baohuan (College of Mathematics and Information Science, Langfang Teachers University, Langfang, Hebei, , P. R. China) Abstract: Let F n+l be the (n + l)-dimensional singular linear space over a finite field F with elements. Denote by L(m, k; n + l, n) the set of all subspaces of type (m 1,k 1)ofF n+l such that 0 k 1 k and 0 m 1 m. If we partially order L(m, k; n + l, n) byordinary inclusion, then L(m, k; n + l, n) is a poset. In this paper, we prove that L(m, k; n + l, n) isa uasi-uniform poset and construct a Leonard pair from L(m, m; n + l, n). Keywords: finite field; singular linear space; uniform poset; Leonard pair MR(2010) Subject Classification: 05B25; 05C50 / CLC number: O157.3 Document code: A Article ID: (2017) Introduction Terwilliger [4] introduced the uniform poset, discussed its algebraic structure and gave eleven families of examples. Woramannotai [10] constructed a family of uniform posets by using the dual polar graphs. Miklavi c and Terwilliger [3] constructed a family of uniform posets by using bipartite Q-polynomial distance-regular graphs. Terwilliger [5] obtained Leonard pairs from classical posets. Their researches stimulate us to consider the uniformity and Leonard pairs of other posets. Let F be a finite field with elements, where is a prime power. Let F n+l be the (n + l)-dimensional singular linear space over F. Denote by L(m, k; n + l, n) thesetofall subspaces of type (m 1,k 1 )off n+l such that 0 k 1 k and 0 m 1 m. If we partially order L(m, k; n + l, n) by ordinary inclusion, then L(m, k; n + l, n) is a poset. In this paper, we prove that L(m, k; n + l, n) is a uasi-uniform poset and construct a Leonard pair from the uniform poset L(m, m; n + l, n). The paper is organized as follows. In Section 1, we recall singular linear spaces, uasi-uniform posets and Leonard pairs. In Section 2, we show that L(m, k; n + l, n) is a uasi-uniform poset. In Section 3, we construct a Leonard pair from L(m, m; n + l, n). 1 Preliminaries In this section we introduce singular linear spaces, uasi-uniform posets and Leonard pairs. Received date: Revised date: Foundation item: The work is supported by NSFC (No ), the NSF of Hebei Province (No. A ), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No ) and the Key Fund Project of Langfang Teachers University (No. LSLZ201403). lizengti@126.com

2 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs Singular Linear Spaces For two non-negative integers n and l, letf n+l denote the (n + l)-dimensional row vector space over F. The set of all (n + l) (n + l) nonsingular matrices over F of the form ( ) T11 T 12, 0 T 22 where T 11 and T 22 are nonsingular n n and l l matrices, respectively, forms a group under matrix multiplication, called the singular general linear group over F and denoted by GL n+l,n (F ). Let P be an m-dimensional subspace of F n+l. Denote also by P an m (n + l) matrixof rank m whose rows span the subspace P and call the matrix P a matrix representation of the subspace P. There is an action of GL n+l,n (F )onf n+l defined as follows, F n+l GL n+l,n (F ) F n+l, ((x 1,x 2,,x n,x n+1,...,x n+l ),T) (x 1,x 2,,x n,x n+1,,x n+l )T. The above action induces an action on the set of subspaces of F n+l ; i.e., a subspace P is carried by T GL n+l,n (F ) to the subspace PT. The vector space F n+l together with the above group action, is called the (n + l)-dimensional singular linear space over F. For 1 i n + l, let e i all other coordinates are 0. be the row vector in F n+l whose i-th coordinate is 1 and Denoted by E the l-dimensional subspace of F n+l generated by e n+1,e n+2,,e n+l. An m-dimensional subspace P of F n+l is called a subspace of type (m, k) if dim(p E) =k. A subspace of type (m, k) is also called (m, k)-subspace. The collection of all the subspaces of types (m, 0) in F n+l,where0 m n, is the attenuated space [9]. For a fixed subspace P of type (m 1,k 1 )inf n+l,letm (m 1,k 1 ; m, k; n + l, n) denotetheset of all the subspaces of type (m, k) containing P, and let N (m 1,k 1 ; m, k; n + l, n) = M (m 1,k 1 ; m, k; n + l, n). By the transitivity of GL n+l,n (F ) on the set of all subspaces of the same type, N (m 1,k 1 ; m, k; n+ l, n) is independent of the particular choice of the subspace P of type (m 1,k 1 ). [9, Lemma 2.3] Lemma 1.1 M (m 1,k 1 ; m, k; n + l, n) is non-empty if and only if 0 k 1 k l and 0 m 1 k 1 m k n hold. Moreover, if M (m 1,k 1 ; m, k; n + l, n) isnon-empty,then [ ] [ ] N (m 1,k 1 ; m, k; n + l, n) = (l k)(m k m1+k1) n (m 1 k 1 ) l k1. (m k) (m 1 k 1 ) k k Quasi-uniform Posets and Incidence Algebras Let P be a poset with partial order. Asusual,wewritea<bwhenever a b and a b. For any two elements a, b P,wesayb covers a, denoted by a b, ifa<band there exists no c P such that a<c<b. Let P be a finite poset with the minimum element, denoted by 0.

3 36 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 By a rank function on P, we mean a function r from P to the set of all the integers such that r(0) = 0 and r(a) =r(b) 1 whenever a b. In this paper, we always assume that P is a poset with the rank function r. The maximum value of r(x) is called the rank of P, denoted by N. Definition 1.1 [4] Let P be a finite poset with N 2. Define the lowering matrices L i, the raising matrices R i and the projection matrices F i (0 i N) inmat P (R) as follows. { 1, if x Pi 1,y P i, x y, (L i ) xy = 0, otherwise, 1 i N, { 1, if x, y Pi,x= y, (F i ) xy = 0 i N. 0, otherwise, Write R i := L t i+1 and R N := 0, L 0 := 0, where Mat P (R) is the set consisting of all P P matrices over the real number field R and whose rows and columns are indexed by P. Definition 1.2 [4] Let P be a finite poset with N 2. The incidence algebra T of P is the real matrix algebra generated by L i,r i,f i (0 i N). Let T be the incidence algebra of the poset P with rank N. Let V be the Euclidean vector space R P over the real number field R with the usual inner product,. We call V the standard module of P. A subspace W of V is called a module of T if it is invariant under T, i.e., tw W for all t T and all w W. A module W of T is said to be irreducible if it is non-zero, and the only non-zero module it contains is W itself. [4, Definition 2.1] Definition 1.3 Let N 2. A parameter matrix of order N is a tridiagonal real matrix E =(e ij ) 1 i,j N satisfying (i) e ii =1(1 i N); (ii) e i,i 1 0(2 i N) ore i,i+1 0(1 i N 1); (iii) the principal submatrix E(r, p) =(e ij ) r+1 i,j p is nonsingular for all integers r, p (0 r p N). We abbreviate e i := e i,i 1 for 2 i N and e + i := e i,i+1 for 1 i N 1. For notational convenience define e 1 := 0 and e+ N := 0. Definition 1.4 Let P be a finite poset of rank N 2 with lowering and raising matrices L i,r i (0 i N). We call P uasi-uniform if there exists a parameter matrix E of order N and three vectors F = {f i } N i=1,h={h i} N i=1,u={u i} N i=1 in RN such that e i R i 2L i 1 L i + L i R i 1 L i + e + i L il i+1 R i = f i L i + h i A + u i B (1 i N), (1.1) where A, B are (0, 1)-matrices in Mat P (R), R 1 = L N+1 =0ande + i,e i are entries in E as indicated in Definition 1.3. Remark 1.1 When H and U are zero vectors, the uasi-uniform poset P is a uniform poset. [4, Theorem 2.5] Lemma 1.2 Let P be a uniform poset of rank N at least 2, with lowering, raising and projection matrices L i,r i,f i for 0 i N. Let T be the incident algebra of P acting on its standard module V.Then (i) V decomposes into an orthogonal direct sum of irreducible T -modules.

4 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs 37 (ii) Each irreducible T -module W has a basis of the form w r,w r+1,,w p (0 r p N) such that (a) w i F i V (r i p), (b) L i w i = w i 1 (r +1 i p) andl r w r =0, (c) R i w i = x i+1 (r, p)w i+1 (r i p 1) and L p w p =0, where x i (r, p) (r +1 i p) are the solution to the system of linear euation x r+1 (r, p) f r+1 x r+2 (r, p) E(r, p).. = f r+2... x p (r, p) f p We refer to the integers r, p in Lemma 1.2 as the endpoints of W. 1.3 Leonard Pairs In this subsection, we recall the Leonard pairs. Throughout this subsection, F denotes an algebraically closed field with characteristic zero. Let X be a suare matrix. Then X is called upper (resp. lower) bidiagonal whenever each nonzero entry lies on either the diagonal or the superdiagonal (resp. subdiagonal). X is called tridiagonal whenever each nonzero entry lies on either the diagonal, subdiagonal, or the superdiagonal. Assume that X is tridiagonal, then X is called irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. Let d denote a nonnegative integer and let Mat d+1 (F) denotethef-algebra consisting of all (d +1) (d + 1) matrices over F. We index the rows and the columns by 0, 1,,d. Let F d+1 denote the F-vector space consisting of all (d +1) 1 matrices over F. Its rows are indexed by 0, 1,,d. We view F d+1 as a left module for Mat d+1 (F). For the rest of the subsection, let V denote a vector space over F that has dimension d +1. LetEnd(V )denotethef-algebra consisting of all linear transformations from V to V. Let {v i } d i=0 denote a basis for V. For X End(V )andy Mat d+1 (F), we say that Y represents X with respect to {v i } d i=0 whenever Xv j = d i=0 Y ijv i for 0 j d. [6, Definition 1.1] Definition 1.5 By a Leonard pair on V, we mean an ordered pair of linear transformations A : V V and A : V V that satisfies the conditions (i) and (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. [7, Definition 22.1] Definition 1.6 By a parameter array over F of diameter d we mean a seuence of scalars ({θ i } d i=0, {θi } d i=0, {ϕ i } d i=1, {φ i } d i=1) taken from F that satisfies the following conditions: (PA1) θ i θ j, θi θ j if i j (0 i, j d).

5 38 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 (PA2) ϕ i 0,φ i 0(1 i d). i 1 θ (PA3) ϕ i = φ h θ d h 1 h=0 (PA4) φ i = ϕ 1 i 1 h=0 θ 0 θ d θ h θ d h θ 0 θ d +(θ i θ 0)(θ i 1 θ d ) (1 i d). +(θi θ 0 )(θ d i+1 θ 0 ) (1 i d). (PA5) The expressions θi 2 θi+1 θ i 1 θ i and θ i 2 θ i+1 θi 1 θ i d 1. [6, Corollary 14.2] Lemma 1.3 Let A = are eual and independent of i for 2 i A and A denote any matrices in Mat d+1 (F) oftheform θ 0 0 θ 0 ϕ θ 1 θ1 ϕ 2 1 θ 2, A = θ ϕd 0 1 θ d 0 θd Then the following are euivalent: (i) (A, A ) is a Leonard pair. (ii) There exist scalars φ 1,φ 2,, φ d in F such that the seuence ({θ i } d i=0, {θ i }d i=0, {ϕ i} d i=1, {φ i } d i=1 ) is a parameter array. 2 The Quasi-uniform Poset L(m, k; n + l, n) In this section, we consider the poset L(m, k; n + l, n) which has the uasi-uniform structure. Let N denote the set of all nonnegative integers. For any x L(m, k; n + l, n), define r(x) =dim(x). Then r: L(m, k; n + l, n) N is the rank function of the poset L(m, k; n + l, n). We remark that the rank of the poset L(m, k; n + l, n) ism. Definition 2.1 Let 2 k<l,0 m k n. Wewrite and L i (m, k; n + l, n) ={x L(m, k; n + l, n) r(x) =i} L(m, k; n + l, n) = m L i (m, k; n + l, n). i=0 For 0 i m, 0 j k, denotebyl j i (m, k; n + l, n) the set of all subspaces of Type (i, j) in L(m, k; n + l, n). Theorem 2.1 The finite poset L(m, k; n + l, n) ofrankm is a uasi-uniform poset. To prove Theorem 2.1, we first give some lemmas. Lemma 2.1 For all x, y L(m, k; n + l, n) with1 m n + l 1, let x and y be the subspaces of type (m 1,k 1 )and(m 2,k 2 ), respectively. If y covers x, i.e., x y, thenm 2 =1+m 1 and k 2 k 1 =0or1. Proof Routine. Lemma 2.2 With reference to Definitions 1.2 and 2.1, let x L i 1 (m, k; n + l, n) and y L i (m, k; n + l, n). Then the following (i) and (ii) hold.

6 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs 39 (i) If x y, then (ii) If x y, then (R i 2 L i 1 L i ) xy = (i 1 1)( +1), 1 i m. { +1, if x y x, (R i 2 L i 1 L i ) xy = 0, otherwise, 2 i m. Proof For a fixed x L i 1 (m, k; n + l, n) andafixedy L i (m, k; n + l, n), we first have (R i 2 L i 1 L i ) xy = (R i 2 ) xz (L i 1 ) zw (L i ) wy z,w L(m,k;n+l,n) = {(z,w) z x, z w, w y}. Next we count the number of the pairs (z,w) such that z x, z w, w y. By Lemma 2.1, we find that z L i 2 (m, k; n + l, n) andw L i 1 (m, k; n + l, n). (i) The case x y. By Definition 2.1 and [8, Corollary 1.8], the number of subspaces z such that z x is i 1 1, and by [8, Corollary 1.9], for each of z there exist + 1 subspaces w such that z w y. Hence, the number of the pairs (z,w) such that z x, z w, w y is ( i 1 1)(). (ii) The case x y. Whenx y x, it follows from z x, z w, w y that x y = z. The number of the pairs (z,w) such that z x, z w, w y is eual to the number of subspaces w such that z w y, whichis +1. Whenx y x, there does not exist z and w such that z x, z w, w y. Lemma 2.3 With reference to Definitions 1.1 and 2.1, let x L i 1 (m, k; n + l, n) and y L i (m, k; n + l, n). Then the following (i) and (ii) hold. (i) If x y, then (L i R i 1 L i ) xy = (ii) If x y, then n+l i+1 + i, if x/ L k i 1 (m, k; n + l, n), 1 i m. n+l i+1 + i l k, if x L k i 1 (m, k; n + l, n), { +1, if y x + y, (L i R i 1 L i ) xy = 0, otherwise, 2 i m k. Proof For a fixed x L i 1 (m, k; n + l, n) andafixedy L i (m, k; n + l, n), we first have (L i R i 1 L i ) xy = (L i ) xz (R i 1 ) zw (L i ) wy z,w L(m,k;n+l,n) = {(z,w) x z,w z,w y}.

7 40 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Next we count the number of the pairs (z,w) such that x z, w z, w y. By Lemma 2.1, we find that z L i (m, k; n + l, n) andw L i 1 (m, k; n + l, n). (i) The case x y. When x / L k i 1 (m, k; n + l, n), if x = w, by Definition 2.1 and [8, Corollary 1.9], the number of subspaces z such that x z is n+l i+1 1. If x w, it implies z = y. By Definition 2.1 and [8, Corollary 1.8], the number of subspaces w such that x w y is i 1 1. Hence, the number of the pairs (z,w) such that x z,w z,w y is n+l i i 1 1=n+l i+1 + i. When x L k i 1 (m, k; n + l, n), if x = w, by Definition 2.1 and Lemma 1.1, the number of subspaces z such that x z is n+l i+1 l k. If x w, it implies z = y. By Definition 2.1 and [8, Corollary 1.8], the number of subspaces w such that x w y is i 1 1. Hence, the number of the pairs (z,w) such that x z,w z,w y is n+l i+1 l k + i 1 1=n+l i+1 + i l k. (ii) The case x y. When y x + y, it follows from x z,w z,w y that y z = w. Thus dim(z + y) =dim(z)+dim(y) dim(w) =i + 1. It implies that z z + y and dim(x + y) = dim(z + y). It follows from x z that x + y = z + y and z x + y. Sincew = y z, wehavethe number of the pairs (z,w) such that x z,w z,w y is eual to the number of subspaces z such that x z x + y, whichis +1. When y x + y, there does not exist z and w such that x z,w z,w y. Lemma 2.4 With reference to Definitions 1.1 and 2.1, let x L i 1 (m, k; n + l, n) and y L i (m, k; n + l, n). Then the following (i) (iii) hold. (i) If x y, then ( n+l i 1)( +1), if y/ L k i (m, k; n + l, n), (L i L i+1 R i ) xy = 1 i m 1. ( n+l i l )( +1), if y L k i (m, k; n + l, n), (ii) If x y, then { +1, if y x + y, (L i L i+1 R i ) xy = 0, if y x + y, 1 i m k 1. (iii) If i = m k, then(l i L i+1 R i ) xy =0. Proof For a fixed x L i 1 (m, k; n + l, n) andafixedy L i (m, k; n + l, n), we first have (L i L i+1 R i ) xy = (L i ) xz (L i+1 ) zw (R i ) wy z,w L(m,k;n+l,n) = {(z,w) x z,z w, y w}.

8 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs 41 Next we count the number of the pairs (z,w) such that x z, z w, y w. By Lemma 2.1, we find that z L i (m, k; n + l, n) andw L i+1 (m, k; n + l, n). (i) The case x y. When y / L k i (m, k; n + l, n), by Definition 2.1 and [8, Corollary 1.9], the number of subspaces w such that y w is n+l i 1. By [8, Corollary 1.8], for each of w there exist + 1 subspaces z such that x z w. Hence, the number of the pairs (z,w) such that x z,z w, y w is (n+l i 1)(). When y L k i (m, k; n + l, n), by Lemma 1.1, the number of subspaces w such that y w is n+l i l k. By [8, Corollary 1.8], for each of w there exist + 1 subspaces z such that x z w. Hence, the number of the pairs (z,w) such that x z,z w, y w is (n+l i l k )(). (ii) The case x y. When y x + y, for1 i m 1, clearly y z. It follows from x z,z w, y w that y + z w. Since i = dim(y) < dim(y + z) dim(w) =i +1, we have dim(y + z) =i +1. So y + z = w, and hence z y + z. Since y x + y, we have dim(x + y) =i +1. It follows from x z that x + y = z + y = w. So the number of the pairs (z,w) such that x z,z w, y w is eual to the number of subspaces z such that x z x + y, whichis +1. Wheny x + y, there does not exist z and w such that x z,z w, y w. (iii) The case i = m. WehaveL i+1 =0,andso(L i L i+1 R i ) xy =0. Lemma 2.5 With reference to Definition 2.1, if x L i 1 (m, k; n + l, n), y L i (m, k; n+l, n) for2 i m, andx y, then x y x if and only if y x + y. Proof Routine. Proof of Theorem 2.1 Define the matrices A and B by means of Definition 1.4 as follows: { 1, if x y, x L k A xy = i 1 (m, k; n + l, n) andy L k i (m, k; n + l, n), 0, otherwise and { 1, if x y, x L k 1 i 1 B xy = (m, k; n + l, n) andy Lk i (m, k; n + l, n), 0, otherwise. To prove the result, we divide the integer i into three cases according to the value of i. Case (I): i =1. By Lemmas , we have the following Table 1: Table 1 Comments x y (R 1L 0L 1) xy 0 (L 1R 0L 1) n+l 1 xy ( (L 1L 2R 1) n+l 1 1)() xy (L 1) xy 1

9 42 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 By Table 1, we obtain the following euation: n+l 1 + (n+l 1 1)( +1) e + 1 = f 1 + h 1 + u 1. Set h 1 = u 1 =0,e 1 = ( +1) 1 and e + 1 = ( +1) 1. Then f 1 = n+l 1. Case (II): 2 i m 1. Note that x y, x y x if and only if y x + y by Lemma 2.5. By Lemmas , we have the following Table 2: Comments x y, x/ L k i 1 (m,k;n+l,n), y/ L k i (m,k;n+l,n) ( (R i 2L i 1L i) i 1 1)() xy (L ir i 1L i) n+l i+1 + i xy ( (L il i+1r i) n+l i 1)() xy Table 2 x y, x L k i 1 (m,k;n+l,n), y L k i (m,k;n+l,n) ( i 1 1)() n+l i+1 + i l k ( n+l i l )() x y, x L k 1 i 1 (m,k;n+l,n), y L k i (m,k;n+l,n) ( i 1 1)() x y, x y x +1 n+l i+1 + i +1 ( n+l i l )() +1 (L i) xy By Table 2, we obtain the following system of linear euations. ( i 1 1)( +1) e i + n+l i+1 + i + (n+l i 1)( +1) e + i = f i, ( i 1 1)( +1) e i + n+l i+1 + i l k + (n+l i l )( +1) e + i = f i + h i, ( i 1 1)( +1) e i + n+l i+1 + i + (n+l i l )( +1) e + i = f i + u i, ( +1)e i +( +1)+( +1)e + i =0. Clearly, (h i,u i,e i,e+ i,f i)=( l l k, l 1, ( +1) 1, ( +1) 1, n+l i ) is the solution of the above system of linear euations, where 2 i m 1. Case (III): i = m. Note that x y, x y x if and only if y x + y by Lemma 2.5. By Lemmas , we have the following Table 3: Comments x y, x/ L k m 1 (m,k;n+l,n) (R m 2L m 1L m) xy ( m 1 1)() (L mr m 1L m) xy n+l m+1 + m Table 3 x y, x L k m 1 (m,k;n+l,n),y Lk m (m,k;n+l,n) ( m 1 1)() x y, x y x +1 n+l m+1 + m l k +1 (L ml m+1r m) xy (L m) xy By Table 3, we obtain the following system of linear euations: ( m 1 1)( +1) e k + n+l m+1 + m = f m + u m, ( m 1 1)( +1) e k + n+l m+1 + m l k = f m + h m, ( +1)e m +( +1)=0.

10 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs 43 Clearly, (h m,u m,e m,e + m,f m )= ( l k 1, 0, ( +1) 1, ( +1) 1, n+l m+1 n+l m + l ) 1 is the solution of the above system of linear euations. Therefore, for 1 i m, wehave e i R i 2L i 1 L i + L i R i 1 L i + e + i L il i+1 R i = f i L i + h i A + u i B. Let E := (e ij ) 1 i,j m = (2.1) and F := (f i ) 1 i m. (2.2) Then by the above arguments, (1.1) holds. To complete the proof, we show that the above matrix E is a parameter matrix. The conditions (i) and (ii) of Definition 1.3 hold obviously. To verify the condition (iii) of Definition 1.3, we calculate the principal minor determinant det E(r, p) := (e ij ) r+1 i,j p, where 0 r<p m. If p m 1, we have ( p r+1 ) 1 det E(r, p) = ( +1) r p 0. (2.3) If p = m, we have det E(r, p) = p r 1 ( +1) r p+1 0. (2.4) So the condition (iii) of Definition 1.3 holds. Therefore, Theorem 2.1 holds. By Remark 1.1 and the proof of Theorem 2.1, we obtain the following remark. Remark 2.1 The poset L(m, m; n + l, n) is a uniform poset. 3 Leonard Pairs From the Uniform Poset L(m, m; n + l, n) In this section, we construct a Leonard pair from the uniform poset L(m, m; n + l, n). Throughout this section, C denotes the field of complex numbers. Lemma 3.1 Let E and F be the matrices in (2.1) and (2.2), respectively. Let E(r, p) :=(e ij ) r+1 i,j p and F (r, p) :=(f r+1,f r+2,,f p ) T.

11 44 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Then the solution to the system of linear euations x r+1 (r, p) x r+2 (r, p) E(r, p) = F (r, p) (3.1). x p (r, p) is x i (r, p) = n+l p r i+1 ( i r )( p i 1 ) () 2, if p m 1, ( i r )( n+l r i 1 ) () 2 i 1, if p = m, where r +1 i p. Proof Obviously, det E(r, p) 0 by (2.3) and (2.4). By calculation, the result holds. For the rest of this section, let V = CL(m, m; n + l, n) be the standard module of L(m, m; n + l, n), and let L i,r i and F i (0 i m) be the lowering, raising and projection matrices, respectively. For convenience, we set Obviously, we have L = m L i and R = i=0 m R i. i=0 F i F j = δ ij F i, 0 i, j m, F 0 + F F m = I. Moreover, F i V = Span{x L(m, m; n + l, n) r(x) =i}, 0 i m. By (1.1), (2.1), (2.2) and Remark 1.1, we get that +1 RL2 LRL L2 R + n+l i L (3.2) vanishes on F i V for 1 i m 1, and vanishes on F m V. We put RL 2 LRL L2 R + n+l m+1 m 1 L (3.3) A = R + m i=0 A = α L + i F i, (3.4) m i=0 i F i, (3.5) where α is any scalar C that is not one of (n+l), (n+l)+1,, (n+l)+m 2. Let T denote the subalgebra of Mat L(m,m;n+l,n) (C) generated by R, L, F 0,F 1,,F m.obviously, R T = L, andeachoff 0,F 1,,F m is symmetric. It follows that T is closed under

12 No. 1 Li Z. T. and Zhang B. H.: Quasi-uniform Posets and Leonard Pairs 45 the conjugate-transpose map. Thus, T is semi-simple, and V is a direct sum of irreducible T -submodules. Theorem 3.1 Let V = CL(m, m; n + l, n) be the standard module of L(m, m; n + l, n) and let W be an irreducible T -submodule of V with endpoints r, p, where0 r<p m 1. Then (A W,A W ) is a Leonard pair on W. Proof We first have AW W, A W W, since the matrices A and A are contained in T by (3.4) and (3.5). Next we show that (A W,A W ) is a Leonard pair on W. Similar to the proof of Lemma 1.2 (ii), by using (3.2) and (3.3) we can show that there exists abasisw r,w r+1,, w p for W such that (i) w i F i V (r i p), (ii) R i w i = w i+1 (r i p 1) and R p w p =0, (iii) L i w i = x i (r, p)w i 1 (r +1 i p) andl r w r =0, where x i (r, p) (r +1 i p) are the solution of the system of linear euations (3.1). Let B (resp. B ) denote the matrix representing A (resp. A ) with respect to the basis w r,w r+1,, w p. Then by the above arguments, we have θ 0 0 θ 0 ϕ θ 1 θ1 ϕ 2 B = 1 θ 2, B = θ... 2,... ϕd 0 1 θ d 0 θd respectively, where d = p r, θ i = r+i, θ i = r i, 0 i d, (3.6) ϕ i = α x r+i (r, p), 1 i d. (3.7) Here α is the scalar from (3.5) and x r+i (r, p) (1 i d) are from Lemma 3.1. Set i 1 θ h θ d h φ i = ϕ 1 +(θi θ 0 θ θ 0 )(θ d i+1 θ 0 ), 1 i d. d h=0 By (3.6) and (3.7), we obtain φ i = (1 i )(1 d i+1 )(1 α n+l+i r d ) () 2 i, 1 i d. (3.8) One readily checks that the above scalars θ i,θi,ϕ i,φ i satisfy the conditions (PA1), (PA2), (PA4) and (PA5) of Definition 1.6. To verify the euation (PA3) of Definition 1.6, we calculate

13 46 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 the right hand of the euation (PA3) of Definition 1.6. Using (3.6), (3.8) and Lemma 3.1, we have i 1 θ h θ d h φ 1 +(θi θ θ 0 θ 0)(θ i 1 θ d ) d h=0 = α n+l d r (1 i )(1 d i+1 ) () 2 = α x r+i (r, p), 1 i d. By (3.7), the (PA3) of Definition 1.6 holds. From the above arguments and by Lemma 1.3, (B,B ) is a Leonard pair in Mat d+1 (C). It follows that (A W,A W ) is also a Leonard pair on W. Acknowledgements The authors thank the referees for their many valuable comments and suggestions. References [1] Aigner, M., Combinatorial Theory, Berlin: Springer-Verlag, [2] Birkhoff, G., Lattice Theory, Providence, RI: AMS, [3] Miklavi c, S. and Terwilliger, P., Bipartite Q-polynomial distance-regular graphs and uniform posets, J. Algebraic Combin., 2013, 38(2): [4] Terwilliger, P., The incidence algebra of a uniform poset, In: Coding Theory and Design Theory, The IMA Volumes in Mathematics and Its Applications, Vol. 20, New York: Springer-Verlag, 1990, [5] Terwilliger, P., An introduction to Leonard pairs and Leonard systems, In: Algebraic Combinatorics (Suzuki, H. ed.), RIMS Kôkyûroku, No. 1109, Kyoto: Kyoto Univ., 1999, [6] Terwilliger, P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl., 2001, 330(1/2/3): [7] Terwilliger, P., An algebraic approach to the Askey scheme of orthogonal polynomials, In: Orthogonal Polynomials and Special Functions: Computation and Applications, Lecture Notes in Math., Vol. 1883, Berlin: Springer-Verlag, 2006, [8] Wan, Z.X., Geometry of Classical Groups over Finite Fields, Beijing: Science Press, [9] Wang, K.S., Guo, J.G. and Li, F., Singular linear space and its applications, Finite Fields Appl., 2011, 17(5): [10] Worawannotai, C., Dual polar graphs, the uantum algebra U (sl 2 ), and Leonard systems of dual -Krawtchouk type, Linear Algebra Appl., 2013, 438(1): &/%$-')ψffi(ff!ο",1#*flfi Leonard ß 465, 723 (ΨΞfiΛ»fl fflφ», ΨΞ, Π, ) 0+ X F n+l Zh`i F W= (n + l)- TfacIH. L L(m, k; n + l, n) ;Y9 E F n+l m=[hmo 0 k 1 k, 0 m 1 m = (m 1,k 1 ) bnih=gf. VD^O89 EB_C> L(m, k; n + l, n) W=SdB_, QN L(m, k; n + l, n) Ze@SdG. :]kpk L(m, k; n + l, n) Ze@RelSdG<UJg L(m, m; n + l, n) AjKe@ Leonard?. ρχν h`i; TfacIH; elsdg; Leonard?