ISSN 2148-838X J. Algebra Comb. Discrete Appl. -(-) • 1–15 Received: 29 March 2022 Accepted: 29 August 2022 AR TI CL E IN PR ES S Journal of Algebra Combinatorics Discrete Structures and Applications Walled Klein-4 Brauer algebras Research Article Annamalai Tamilselvi, Dhilshath Shajahan Abstract: The new class of diagram algebras known as walled Klein-4 Brauer algebras, denoted by −→−→ Dr,s(l), where r, s ∈ N and l ∈ K, is an indeterminate, is studied in this paper. The walled klein-4 Brauer algebras are explained in terms of generators and relations. The indexing set of the simple modules of the walled Klein-4 Brauer algebras was described. We established that walled Klein-4 Brauer algebras are iterated inflations of the group algebra of the group (Z2 ×Z2) oSr ×(Z2 ×Z2) oSs, and we concluded that −→−→ Dr,s(l) is cellular as a consequence. 2010 MSC: 16G30, 05E10, 20C30 Keywords: Cellularity, Walled Brauer algebras, Walled signed Brauer algebras 1. Introduction Richard Brauer introduced the Brauer algebra Bn(δ) in the representation theory of orthogonal groups. Undirected diagrams are the basis elements of Brauer algebra. The centralizers of the orthogonal groups are these algebras. G-Brauer algebras were introduced by Parvathi and Savithri, with signed diagram as basis and studied in [7] its structure over k(x), where x is an indeterminate. The symmetric group Sr and the general linear group GLn(C) representation theories over C are linked through Schur-Weyl duality. The walled Brauer algebras Br,s(δ) is the result of a third version of Schur-Weyl duality. This algebra was examined by Turaev, Koike, and Benkart et al. Kethesan introduced a new class of diagram algebras called walled signed Brauer algebras denoted by −→ Dr,s(x), where x is an indeterminate. The fr,s map gives a vector space isomorphism between the group algebra of the hyperoctahedral group Z2 o Sr+s and the walled singed Brauer algebras −→ Dr,s(x). The structure of walled signed Brauer algebras has been studied in [4]. In [11], Tamilselvi et al., studied Annamalai Tamilselvi (Corresponding Author); Ramanujan Institute for Advanced Study in Mathematics, Uni- versity of Madras, Chennai, India (email: tamilselvi.riasm@gmail.com). Dhilshath Shajahan; Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India (email: dhilshajahan@gmail.com). 1 https://orcid.org/0000-0002-0682-2705 https://orcid.org/0000-0002-1448-0792 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 the Robinson-Schensted correspondence for the walled Brauer algebras and the walled signed Brauer algebras. Cellularity and representations of walled cyclic G-Brauer algebras were studied by Tamilselvi [10]. In [8], Parvathi and Sivakumar studied about a new class of diagram algebras called Klein-4 diagarm algebras denoted by Rk(n). These algebras are the centralizer algebras of the group (Z2 ×Z2) oSn acting on V ⊗k, where V is the signed permutation module for (Z2 ×Z2) o Sn. In that paper [8], they give a combinatorial rule for the decomposition of the tensor powers of the signed permutation representation of (Z2 ×Z2) oSn, by explicitly constructing the basis for the irreducible modules. We were motivated by this work to create a new class of algebras over K, known as walled Klein-4 Brauer algebras and denoted by −→−→ Dr,s(l), where l ∈ K is an indeterminate. In section 3, we define walled Klein-4 Brauer algebras and show how they are represented in terms of generators and relations. We define the map Flipr,s, which is a vector space isomorphism between the group algebra of the group (Z2 ×Z2) o Sr+s and the walled Klein-4 Brauer algebras, in that section. Using the representations of the group algebra of the group [8, 9], we described the indexing set of the simple modules of the walled Klein-4 Brauer algebras in section 4. In section 5, we proved that the walled Klein-4 Brauer algebras are the iterated inflations of the group algebra of the group (Z2 ×Z2) oSr×(Z2 ×Z2) oSs, and we concluded that walled Klein-4 diagram algebras are cellular. 2. Preliminaries Now, we will present some definitions that we will use to proceed on. Definition 2.1. [3] A partition β = (β1,β2, . . . ,βt) of (r + s) is said to be p-regular if either p > 0 and there is no 1 ≤ i ≤ t such that βi = βi+1 = . . . = βi+p or p = 0. Definition 2.2. ([9]) A 4-partition ((β1), (β2), (β3), (β4)) of size (r+s), is an ordered 4-tuple of partitions (β1), (β2), (β3) and (β4) such that |(β1), (β2), (β3), (β4)| = |(β1)| + |(β2)| + |(β3)| + |(β4)| = (r + s). , , , The 4-partition ((β1), (β2), (β3), (β4)) = ([2, 2][3, 2][2, 1][12]) is represented by the above 4-tableau and the sum |(β1)| + |(β2)| + |(β3)| + |(β4)| = 14. The set of all 4-partitions of (r + s) is denoted by B4(r+s). Definition 2.3. If the partitions (β1), (β2), (β3) and (β4) are p-regular then the 4-partition ((β1), (β2), (β3), (β4)) of (r + s) is said to be p-regular. Theorem 2.4. ([2], theorem (10.33)) Let K be a field which is a splitting field for the finite groups G1 and G2. Then: • For each simple KG1 module M and each simple KG2 module N, the outer tensor product M �N is a simple K(G1 ×G2)-module. • Every simple K(G1 × G2)-module is of the above form M � N, with M and N are uniquely determined (up to isomorphism). Remark 2.5. [8] A finite dimensional associative algebra A with unit over C, the field of complex numbers is said to be semi-simple if A is isomorphic to a direct sum of full matrix algebras A∼= ⊕ λ∈ Mdλ(C) for  a finite index set, and dλ positive integers. Corresponding to each λ ∈ Â, there is a single irreducible A module call it V λ which has dimension dλ. If  is a singleton set then A is said to be simple. Maschke’s theorem says that for G finite, C[G] is semi-simple. 2 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Remark 2.6. [5] An algebra is cellular if an only if it can be written as the iterated inflations of copies of full matrix algebras. Moreover iterated inflation of cellular algebras is always cellular again ([5], proposition 3.4). 3. Walled Klein-4 Brauer algebras In this section, we define the walled Klein-4 Brauer algebras which are subalgebras of the G-Brauer algebras [7], where G is the Klein-4 group and give a presentation of walled Klein-4 Brauer algebras in terms of generators and relations. Throughout this paper we are considering 1 as the identity. Fix an algebraically closed field K of characteristic p ≥ 0 and l an indeterminate. For r,s ∈ N the walled Klein-4 Brauer algebras −→−→ Dr,s(l) can be defined in the following manner. A diagram is said to be signed diagram if every edge is labeled by a sign. In the walled Klein-4 Brauer algebras we are considering four types of signs (e,e), (e,b), (a,e) and (a,b). The Klein-4 Brauer algebra −→−→ Dr+s (diagrams without wall) which is the G-Brauer algebra defined in [7], is the set of all signed diagrams −→−→ d with n signed edges and 2n vertices arranged in two rows of n vertices each. Each signed edge links to exactly two vertices in these signed diagrams, and each vertex links to exactly one signed edge. Edges connecting the top row vertex and the bottom row vertex are called vertical edges, and the remaining edges are called upper (connecting two vertices on the top row) or lower (connecting two veritices on the bottom row) horizontal curves (edges connecting two vertices in the same row). 3.1. Multiplication of walled Klein-4 Brauer diagrams To find the product of two signed diagrams −→−→ d 1 and −→−→ d 2, draw −→−→ d 2 below −→−→ d 1 and connect the ith upper vertex of −→−→ d 2 with the ith lower vertex of −→−→ d 1. A new edge obtained in the product −→−→ d 1. −→−→ d 2 is labeled by a sign (e,e) or (e,b) or (a,e) or (a,b) according as the product of the signs (coordinate wise) of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this edge, where the product of the coordinates of the signs defined to be e = e.e = a.a = b.b; a = e.a = a.e; b = e.b = b.e. The product of the diagrams produces a new diagram with closed curves called loops. A loop α in −→−→ d 1. −→−→ d 2 is replaced by the variable l2, if the product of signs of the edges obtained from−→−→ d 1 and −→−→ d 2 to form this loop is (e,e) or (a,b). Similary the loop α in −→−→ d 1. −→−→ d 2 is replaced by the variable l, if the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this loop is (a,e) or (e,b). Now we have −→−→c , which is a signed diagram with each edge is marked as above and −→−→ d 1. −→−→ d 2 = l 2e1+e2. −→−→c , where e1 is the number of loops with the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form those loops is (e,e) or (a,b), e2 is the number of loops with the product of signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form those loops is (e,b) or (a,e). It is usual to represent basis elements by diagrams with (r + s) vertices on the top row numbered 1 to (r + s) from left to right and (r + s) vertices on the bottom row numbered 1 to (r + s) from left to right, where each vertex is connected to precisely one other by a signed edge. We now define the walled Klein-4 Brauer algebra when n = r + s. Partition the basis diagrams of−→−→ Dr+s with the wall separating the first r vertices in the top row and the first r vertices in the bottom row from the remaining s vertices in the top row and s vertices in the bottom row, obtained the walled 3 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Klein-4 Brauer algebras −→−→ Dr,s(l) which is the diagram algebra with basis of those signed diagrams with no vertical edges crosses the wall and every upper or lower horizontal curves does cross the wall. The example of the basis elements −→−→ d 1 and −→−→ d 2 in −→−→ D3,2(l), and the multiplication of −→−→ d 1 and −→−→ d 2 are given below, −→−→ d 1 = 1 2 3 4 5 1 2 3 4 5 (a, e) (e, b) (e, e) (e, e) (e, e) −→−→ d 2 = 1 2 3 4 5 1 2 3 4 5 (a, e) (a, b) (e, b) (e, e) (e, e) 1 1 2 3 4 5 1 2 3 4 5 (a, e) (e, b) (e, e) (e, e) (e, e) 1 2 3 4 5 1 2 3 4 5 (a, e) (a, b) (e, b) (e, e) (e, e) −→−→ d 1. −→−→ d 2 = = l 2 1 2 3 4 5 1 2 3 4 5 (e, b)(a, e) (e, e) (e, e) (e, e) 1 The edge (1, 5) in the resulting diagram is labeled by the sign (e,b), since the coordinate wise product of the signs of the edges obtained from −→−→ d 1 and −→−→ d 2 to form this edge is (a,e).(a,b).(e,e) = (a.a.e,e.b.e) = (e,b). Similarly the other edges are labeled by the corresponding signs in the resulting diagram. Also the resulting diagram is multiplied with l2 since the product of the signs of the edges from −→−→ d 1 and −→−→ d 2 to form the loop in the product −→−→ d 1. −→−→ d 2 is (e,b).(e,b) = (e.e,b.b) = (e,e). It is useful to compare −→−→ Dr,s(l) with the group algebra K((Z2 × Z2) o Sr+s) of the group G = (Z2 ×Z2) oSr+s; where Sr+s is a symmetric group of (r + s) symbols and (Z2 ×Z2) is the Klein-4 group consisting of four elements. The group algebra can be viewed diagrammatically if we identify the element g of G with its signed diagram with no horizontal edge. We define a map, Flipr,s : K((Z2 ×Z2) o Sr+s) → −→−→ Dr,s(l), mapping a Klein-4 signed diagram with no horizontal curves to the walled Klein-4 signed diagram obtained by adding a wall between the rthand (r + 1)th vertices, then without disconnecting any edges or changing the sign flip the portion of the diagram that is already on the right side of the wall. For example, 1 2 3 4 5 1 2 3 4 5 (e, e) (e, e) −→−→ f = (e, a) (e, b) (a, b) 1 2 3 4 5 1 2 3 4 5 (e, e) (e, e) (e, a) (e, b) (a, b) −→−→ h = 1 The walled Klein-4 Brauer diagram −→−→ h above arise by applying Flip3,2 to the Klein-4 signed diagram 4 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 −→−→ f . Once add the wall in −→−→ f between the vertices 3 and 4 flip it on the right side of the wall, so that the edges (3, 4) and (1, 4) in −→−→ f changes to (3, 4) and (1, 4) in −→−→ h respectively. It follows in particular that the map Flipr,s is a vector space isomorphism. since dim −→−→ D r,s (l) = dim K((Z2 ×Z2) oSr+s) = 4r+s.(r + s)! 3.2. Generators and relations of the group algebra K((Z2 ×Z2) oSr+s) [9] The algebra G(r+s) = KG of the group G = (Z2 × Z2) o Sr+s is generated by t1, t2 and the transpositions si := (i, i + 1) for i = 1, 2, ..., (r + s− 1) subject to the following relations • t21 = 1 • t22 = 1 • si2 = 1 for 1 ≤ i ≤ r + s− 1 • t1t2 = t2t1 • t1s1t1s1 = s1t1s1t1 • t2s1t2s1 = s1t2s1t2 • t2s1t1s1 = s1t1s1t2 • sisj = sjsi if |i− j| ≥ 2 • sisi+1si= si+1sisi+1 for 1 ≤ i ≤ r + s− 1 • sit1 = t1si for i ≥ 2 • sit2 = t2si for i ≥ 2 Let Flipr,s(t1) = q1, Flipr,s(t2) = q2, and for i = 1, 2, . . . , (r−1), (r + 1), . . . , (r + s−1), Flipr,s(si) = si(with the addition of the wall). When i = r then Flipr,s(sr) = er is the diagram, 3.3. Generators and relations of the walled Klein-4 Brauer algebras Theorem 3.1. The walled Klein-4 Brauer algebras −→−→ Dr,s(l) is generated by the elements −→−→q 1, −→−→q 1′, −→−→q r+1,−→−→q (r+1)′, s1, s2, ..., sr−1, er, sr+1, ..., sr+s−1 and satisfying the following relations: 1. s2i = 1 for i = 1, 2, ...(r − 1), (r + 1)...(r + s− 1). 2. sisj = sjsi if |i− j| > 1. 3. si+1sisi+1 = sisi+1si for i 6= r,r − 1. 5 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 4. ersi = sier for 1 ≤ i ≤ r − 2 or (r + 2) ≤ i ≤ (r + s− 1). 5. (er)2 = l2er. 6. ersr−1er = er. 7. ersr+1er = er. 8. sr−1sr+1ersr−1sr+1er = ersr−1sr+1er. 9. ersr−1sr+1ersr−1sr+1 = ersr−1sr+1er. 10. −→−→q 2i = 1 for i = 1, 1 ′ , 2, 2 ′ , ...(r + s), (r + s) ′ . 11. −→−→q i sj = sj −→−→q i for i = 1, 1 ′ and j 6= i. 12. −→−→q i s1 −→−→q i s1 = s1 −→−→q i s1 −→−→q i for i = 1, 1′. 13. −→−→q isj = sj −→−→q i for i = (r + 1), (r + 1) ′ and j 6= r, (r + 1). 14. −→−→q isr+1 −→−→q isr+1 = sr+1 −→−→q isr+1 −→−→q i for i = (r + 1), (r + 1) ′ . 15. er −→−→q ier = ler for i = r,r ′ , (r + 1), (r + 1) ′ 16. si −→−→q i+1 = −→−→q isi for i 6= r,r ′ . 17. er −→−→q j = er −→−→q j−1 for j = (r + 1), (r + 1) ′ . 18. −→−→q jer = −→−→q j−1er for j = (r + 1), (r + 1) ′ . 19. er −→−→q i = −→−→q ier for i 6= r,r ′ , (r + 1), (r + 1)′. 20. er −→−→q isr+1er = er −→−→q i+2 for i = r,r′. 21. er −→−→q isr+1er = er −→−→q i+1 for i = r + 1, (r + 1)′. where r′ + i = (r + i)′ 6 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 For 1 ≤ i ≤ r − 1, For i = r, For r + 1 ≤ i ≤ r + s− 1 Proof. The walled Brauer algebra Dr,s(l2) for the symmetric group Sr+s is generated by s1, s2, . . . , sr−1, er, sr+1, . . . , sr+s−1 with the relations from (1) to (9). By the universal property of free algebra any homomorphism from {s1, s2, . . . , sr−1, er, sr+1, . . ., sr+s−1} to B, where B is the associative algebra over K and is generated by p1, p1′, pr+1, p(r+1)′, t1, t2, . . . , tr−1, xr, tr+1, . . . , tr+s−1 satisfies the relations from (1) to (21) can be extended to a homomorphism from Dr,s(l2) to B. So there exist a unique algebra homomorphism, Ψ ′ : Dr,s(l 2) → B such that Ψ′(s1), Ψ ′ (s2), . . . , Ψ ′ (sr−1), Ψ ′ (er), Ψ ′ (sr+1), . . . , Ψ ′ sr+s−1 satisfies the relation from (1) to (9), where Ψ′(si) = ti, for i = 1, 2, . . . , r− 1, r + 1, . . . , r + s− 1 and Ψ′(er) = xr. Similarly the generators −→−→q 1, −→−→q 1′, −→−→q r+1, −→−→q (r+1)′, s1, s2, . . . , sr−1, sr+1, . . . , sr+s−1 satisfies the relations of the groups (Z2 ×Z2) o Sr and (Z2 ×Z2) o Ss that is the relations (1), (2), (3), (10), (11), (12), (13), (14) and (16). Again by the universal property of free algebra any homomorphism from { −→−→q 1, −→−→q 1′, −→−→q r+1, −→−→q (r+1)′, s1, s2, . . . , sr−1, sr+1, . . . , sr+s−1 } to B can be extended to a homomorphism from K((Z2 ×Z2) o Sr × (Z2 ×Z2) oSs) to B. So there exist a unique algebra homomorphism, Ψ′′ : K((Z2 ×Z2) oSr × (Z2 ×Z2) oSs) → B such that Ψ′′( −→−→q 1), Ψ′′( −→−→q 1′), Ψ′′( −→−→q r+1), Ψ′′( −→−→q (r+1)′), Ψ′′(s1), Ψ ′′(s2), . . . , Ψ ′′(sr−1), Ψ ′′(sr+1), . . . , Ψ ′′(sr+s−1) satisfies the relations of the groups (Z2×Z2)oSr and (Z2×Z2)oSs that is the relations (1), (2), (3), (10), (11), (12), (13), (14) and (16). Also Ψ′|K(Sr×Ss) = Ψ′′|K(Sr×Ss) and Ψ′′( −→−→q 1) = p1, Ψ′′( −→−→q 1′) = p1′, Ψ′′( −→−→q r+1) = pr+1, Ψ′′( −→−→q (r+1)′) = p(r+1)′ and Ψ′′(si) = ti. First, for the walled Brauer diagram w ∈ Dr,s(l2), we prove the following 7 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 (i) If w −→−→q i −→−→q j = w, then Ψ′(w)Ψ′′( −→−→q i)Ψ′′( −→−→q j) = Ψ′(w) for 1, 1′ ≤ i ≤ r,r′ and (r + 1), (r + 1)′ ≤ j ≤ (r + s), (r + s)′. (ii) If −→−→q i −→−→q jw = w, then Ψ′′( −→−→q i)Ψ′′( −→−→q j)Ψ′(w) = Ψ′(w) for 1, 1′ ≤ i ≤ r,r′ and (r + 1), (r + 1)′ ≤ j ≤ (r + s), (r + s)′. (iii) If −→−→q sw −→−→q t = w, then Ψ′′( −→−→q s)Ψ′(w)Ψ′′( −→−→q t) = Ψ′(w) for 1, 1′ ≤ s,t ≤ r,r′ and (r + 1), (r + 1)′ ≤ s,t ≤ (r + s), (r + s)′. By extending the calculations for 1′ ≤ i ≤ r′, (r + 1)′ ≤ j ≤ (r + s)′ and considering Ψ′ is an algebra homomorphism on Dr,s(l2), Ψ′′ is an algebra homomorphism on K((Z2 ×Z2) oSr × (Z2 ×Z2) oSs) and Ψ′|K(Sr×Ss) = Ψ′′|K(Sr×Ss)), the proofs of (i), (ii) and (iii) are very similar to the proofs (a), (b) and (c) of the theorem 3.2, [4]. Next, using the results (i), (ii) and (iii), we define the homomorphism from −→−→ Dr,s(l) to B by the following way. If −→−→w ∈ −→−→ Dr,s(l), then there exist −→−→q , −→−→ q′ ∈ H so that −→−→w = −→−→q w −→−→ q′ , with w being the underlying walled Klein-4 Brauer diagram of −→−→w and H being the subgroup of (Z2 ×Z2) oSr × (Z2 ×Z2) oSs generated by−→−→q i, where 1, 1′ ≤ i ≤ (r + s), (r + s)′. If there are other elements −→−→ q′′, −→−→ q′′′ ∈ H and make −→−→w = −→−→ q′′w −→−→ q′′′ then −→−→q w −→−→ q′ = −→−→ q′′w −→−→ q′′′ implies that w = −→−→q −→−→ q′′w −→−→ q′′′ −→−→ q′ . So we get, Ψ′′( −→−→q −→−→ q′′)Ψ′(w)Ψ′′( −→−→ q′′′ −→−→ q′ ) = Ψ′(w) Ψ′′( −→−→q )Ψ′′( −→−→ q′′)Ψ′(w)Ψ′′( −→−→ q′′′)Ψ′′( −→−→ q′ ) = Ψ′(w) Ψ′′( −→−→ q′′)Ψ′(w)(Ψ′′( −→−→ q′′′) = Ψ′′( −→−→q )Ψ′(w)Ψ′′( −→−→ q′ ) Now define Ψ : −→−→ Dr,s(l) → B by Ψ( −→−→w ) = Ψ′′( −→−→q ) Ψ′(w) Ψ′′( −→−→ q′ ). Ψ = Ψ′ on Dr,s(l2), Ψ = Ψ′′ on K((Z2 ×Z2) o Sr × (Z2 ×Z2) o Ss) are obvious. We can use the linearity condition to extend it to the entire space. Again by extending the calculations for 1′ ≤ i ≤ r′, (r + 1)′ ≤ j ≤ (r + s)′, the proof of the following identity, Ψ(eĭ,j̆ −→−→q eŭ,v̆) = Ψ′(eĭ,j̆)Ψ ′′( −→−→q )Ψ′(eŭ,v̆) (1) where, eĭ,j̆ = ∏ p eip,jp, eŭ,v̆ = ∏ q euq,vq, −→−→q = ∏ n −→−→q kn, where 1 ≤ ip,uq ≤ r, (r + 1) ≤ jp,vq ≤ (r + s), 1, 1′ ≤ kn ≤ (r + s), (r + s)′, is similar to the proof of the identity (3.1.1) in theorem 3.2, [4]. Using the identity (1) and the proof of theorem 3.2, [4], we get that Ψ is an isomorphism. 4. Indexing set for the simple modules of the walled Klein-4 Brauer algebras This section will cover the indexing set for the simple modules of the walled Klein-4 Brauer algebra−→−→ Dr,s(l), using the idempotent of that algebra and the simple modules of the group algebra of the groups (Z2 ×Z2) o Sr and (Z2 ×Z2) o Ss. The walled Klein-4 Brauer algebra −→−→ Dr,s(l) is also used to build the recollment towers discussed in [1]. For K an algebraically closed field, r,s > 0 and l 6= 0 the following is a description of an idempotent element of the walled Klein-4 Brauer algebra. Let −→−→e r,s be the walled Klein-4 Brauer diagram multiplied 8 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 by l−2, with one horizontal curve linking r and (r + 1) at the top marked by the sign (e,e), one horizontal curve linking r and (r + 1) at the bottom marked by the sign (e,e). The remaining edges for i = 1, 2, . . .r−1,r + 2, . . . ,s, are vertical edges that link i and i, and they are marked by the sign (e,e). The diagram below demonstrates this. −→−→e r,s is certainly an idempotent in −→−→ Dr,s(l). Because ( −→−→e r,s)2 equals −→−→e r,s. Lemma 4.1. The algebra −→−→ Dr−1,s−1(l) is isomorphic to the algebra −→−→e r,s −→−→ Dr,s(l) −→−→e r,s, for r,s > 0. Proof. For every diagram −→−→ d that belongs to −→−→e r,s −→−→ Dr,s(l) −→−→e r,s, −→−→ d can be constructed from −→−→ Dr−1,s−1(l) by inserting two signed vertical edges (vertical edges marked by the sign (e,e) or (a,e) or (e,b) or (a,b)) before and after the wall in some −→−→ f that belongs to −→−→ Dr−1,s−1(l), so that r is linked with r and (r + 1) is linked with (r + 1). By this way we can define an injective homomorphism between the algebras −→−→ Dr−1,s−1(l) and −→−→e r,s −→−→ Dr,s(l) −→−→e r,s. For any image −→−→ d belongs to −→−→e r,s −→−→ Dr,s(l) −→−→e r,s, we can get a pre-image −→−→ f belongs to −→−→ Dr−1,s−1(l) by removing the vertical edges with the sign that we inserted to get −→−→ d . As a result, the algebras −→−→ Dr−1,s−1(l) and −→−→e r,s −→−→ Dr,s(l) −→−→e r,s are isomorphic. Between the algebras, the preceding lemma aids in defining the necessary exact localization and the right exact globalisation functor [3]. Next we will collect some details about the representations and the simple modules of the group (Z2 ×Z2) oSn from [9]. The collection of all 4-partitions of (r + s), that is B4(r+s) is the indexing set for the inequivalent irreducible representations of the group (Z2×Z2)oSr+s. For each λ ∈ B4(r+s) the set, {πλ: (Z2×Z2)oSr+s → End(Vλ)} is a complete set of inequivalent irreducible representation of the group (Z2 ×Z2) o Sr+s. The set {Vλ: λ ∈ B4(r+s)} is a complete set of simple modules of (Z2 ×Z2) oSr+s. By extending the multiplication turn the representations of the group (Z2×Z2)oSr+s into the group algebra K((Z2 ×Z2) oSr+s) the simple modules of K((Z2 ×Z2) oSr×(Z2 ×Z2) oSs) are the outer tensor product of the simple modules of the form VλL � VλR, where VλL is the simple module of the group algebra K((Z2 ×Z2) o Sr), VλR is the simple module of the group algebra K(Z2 ×Z2) o Ss) and λL ∈ B4r, λR ∈ B4s, λL and λR are the p-regular 4-partition of r and p-regular 4-partition of s respectively [[2], theorem (10.33)]. Consider the indexing set of the simple modules of the group algebra K((Z2×Z2)oSr×(Z2×Z2)oSs) as −→−→ In r,s 4−reg. So we get, −→−→ In r,s 4−reg= {(λL, λR) : λL is a p-regular 4-partition of r and λR is a p-regular 4-partition of s}. If the characteristic p of the field K is zero or p > max(r,s) then the group algebra K((Z2 ×Z2) o Sr × (Z2 ×Z2) oSs) is semisimple [2]. Next, we will give the definition and lemma that we will need to construct the indexing set simple modules of the algebra −→−→ Dr,s(l). 9 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Definition 4.2. The vertical pair of a diagram −→−→ d that belongs to −→−→ Dr,s(l) is an ordered pair (α,β), where α represents the number of signed vertical edges on the left side of the wall and β represents the number of singed vertical edges on the right side of the wall, and the remaining top and bottom vertices are linked in pairs with signed horizontal curves. The edges marked by the signs (e,e) or (a,e) or (e,b) or (a,b) are referred to as signed edges. Remark 4.3. Whenever we multiply the two walled Klein-4 Brauer diagrams −→−→ d and −→−→e with the vertical pairs (α,β) and (γ,δ), we get a walled Klein-4 Brauer diagram with the vertical pair (σ,τ), with σ ≤ min(α,γ) and τ ≤ min(β,δ). Lemma 4.4. (a) The set −→−→ Dr,s(l) −→−→e r,s −→−→ Dr,s(l) has a basis of all diagrams with vertical pair (a,b) for some a ≤ (r − 1) and b ≤ (s− 1). (b) −→−→ Dr,s(l) −→−→e r,s −→−→ Dr,s(l) is an ideal of −→−→ Dr,s(l). (c) The quotient −→−→ Dr,s(l) / −→−→ Dr,s(l) −→−→e r,s −→−→ Dr,s(l) has a basis of all diagrams with vertical pair (r,s). (d) The quotient −→−→ Dr,s(l) / −→−→ Dr,s(l) −→−→e r,s −→−→ Dr,s(l) is isomorphic to the algebra K((Z2 ×Z2) oSr ×(Z2 × Z2) oSs). Proof. The proofs of (a), (b) and (c), follows from the multiplication of the walled Klein-4 Brauer diagrams that we defined in section(3.1), and the definition(4.2), of the vertical pair of the diagram. Again the proof (d), follows from (c). Subsequently, for l = 0, We define an alternate idempotent −̃→−→e r,s in a different way as we cannot define the idempotent element of the walled klein-4 algebra as we did before. If r or s is greater than or equal to 2, then −̃→−→e r,s is the walled Klein-4 Brauer diagram with one upper horizontal curve marked by the sign (e,e) linking r and (r + 1), one lower horizontal curve marked by the sign (e,e) connecting r and (r + 2), nodes (r + 2) and (r + 1) are linked by a vertical edge marked by the sign (e,e), and all other edges being vertical lines from i to i marked by the sign (e,e). This is depicted in the diagram below, −̃→−→e r,s is definitely an idempotent. The lemma(4.1) and lemma(4.4) also applicable to −̃→−→e r,s. Theorem 4.5. The indexing set −→−→ Inr,s of the simple modules of the walled Klein-4 Brauer algebra −→−→ Dr,s(l) equals to the disjoint union of −→−→ In r−i,s−i 4−reg , where 0 ≤ i ≤ min(r,s), for l 6= 0 or r 6= s. Proof. The Isomorphism defined in lemma(4.1) allows us to define an exact localization functor [3], F : −→−→ Dr,s- mod → −→−→ Dr−1,s−1- mod by Lr,s(M)= −→−→e r,s(M); M ∈ −→−→ Dr,s- mod, and a corresponding right exact globalization functor, Gr−1,s−1 : −→−→ Dr−1,s−1- mod → −→−→ Dr,s - mod by, 10 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Gr−1,s−1(N) = −→−→ Dr,s −→−→e r,s ⊗−→−→e r,s −→−→ Dr,s −→−→e r,s (N); N ∈ −→−→ Dr−1,s−1 - mod. By theorem 1 in [1] and the isomorphism (d) in lemma(4.4), we have that for r,s > 0, −→−→ Inr,s = −→−→ Inr−1,s−1 t −→−→ In r,s 4−reg = −→−→ Inr−2,s−2 t −→−→ Inr−1,s−1 t −→−→ In r,s 4−reg and so on, we have −→−→ Inr,s = min(r,s)⊔ i=0 −→−→ In r−i,s−i 4−reg as −→−→ Dr,0 ∼= −→−→ D0,r ∼= K((Z2 ×Z2) oSr) Next we define the sequence of idempotents −→−→e r,s,i in the walled Klein-4 Brauer algebra −→−→ Dr,s (l) to produce the chain of two sided ideals in that algebra. Consider, −→−→e r,s,0 = 1 and −→−→e r,s,i is the isomorphic image of ( −→−→e r−1,s−1,i−1), which we defined in the lemma(4.1), for 1 ≤ i ≤ min(r,s). The element −→−→e r,s,i is undefined when l = 0, i = r, and r = s. To these sequence of idempotents we define associate quotients, −→−→ Dr,s,i = −→−→ Dr,s / −→−→ Dr,s −→−→e r,s,i −→−→ Dr,s. When l 6= 0, we can use our explicit explanation of the isomorphism defined in the lemma 4.1 to provide an equivalent description of −→−→e r,s,i as l−2i times the walled Klein-4 Brauer diagram with i upper horizontal curves marked by the sign (e,e) linking r− t to r + 1 + t, j lower horizontal curves marked by the sign (e,e) linking r − t to r + 1 + t for 0 ≤ t ≤ i−1, and the remaining edges are all vertical marked by the sign (e,e) and linking i to i. In the case of l = 0, a similar justification can be provided. When i = 2 and l 6= 0, the idempotent −→−→e r,s,2 is shown in the diagram below. Let −→−→ Ai = −→−→ Dr,s −→−→e r,s,i −→−→ Dr,s then, −→−→ A0 = −→−→ Dr,s −→−→e r,s,0 −→−→ Dr,s = −→−→ Dr,s , −→−→ A1 = −→−→ Dr,s −→−→e r,s,1 −→−→ Dr,s ⊂ −→−→ Dr,s and so on. So we get the sequence of ideals −→−→ Ai such that, . . . ⊂ −→−→ Ai+1 ⊂ −→−→ Ai ⊂ −→−→ Ai−1 ⊂ . . . ⊂ −→−→ A1 ⊂ −→−→ A0 = −→−→ Dr,s (2) Remark 4.6. The ideal −→−→ Aj, for j = 1, 2, 3, . . . , i−1, i, i + 1, . . . , has a basis of all diagrams with veritcal pair −−−→−−−→ (α,β) for some α ≤ r − j and β ≤ s− j. In particular the quotient −→−→ Aj / −→−→ Aj+1 for j = 1, 2, 3, . . . , i−1, i, i + 1, . . . , has a basis of all diagrams with vertical pair −−−−−−−−−→−−−−−−−−−→ (r − j,s− j). 11 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 5. Walled Klein-4 Brauer algebras are iterated inflations We show that walled Klein-4 Brauer algebras are iterated inflations [5], of group algebra of the group ((Z2×Z2)oSr×(Z2×Z2)oSs) in this section. As a result, we will see that walled Klein-4 Brauer algebras are cellular, and we can also see that the simple modules of the walled Klein-4 Brauer algebra over a field K are described. We will start by proving some results for constructing inflated algebras. 5.1. One row representation of Walled Klein-4 Brauer diagram Each of the Walled Klein-4 Brauer diagram −→−→ d ∈ −→−→ Dr,s with f top signed horizontal curves, f bottom signed horizontal curves and the remaining edges are being vertical can be represented uniquely by a partial one row diagram such that −→−→ d = Y−→−→µ, −→−→ν ,ρ(µν) . Here −→−→µ stands for the top signed horizontal curves configuration, −→−→ν for the bottom signed horizontal curves configuration and ρ(µν) for the remaining signed vertical edge configuration obtained by renumbering the top vertices of the signed vertical edges from left to right as 1, 2, . . . ,r−f,r−f + 1,r−f + 2, . . . ,r + s−2f and their bottom vertices from left to right as 1̄, 2̄, . . . ,r −f,r −f + 1,r −f + 2, . . . ,r + s− 2f. Also ρ(µν) = (ρ,ζ) ∈ (Z2×Z2)oSr−f×(Z2×Z2)oSs−f such that ρ(i) = j̄ if the ith top vertex of the the vertical edge is linked to the bottom vertex j̄, where ρ ∈ Sr−f ×Ss−f and ζ is a map from {1, 2, . . . ,r−f,r−f + 1,r−f + 2, . . . ,r + s−2f}−→ Z2 ×Z2 with, ζ(i) =   (e,e), if the sign (e,e) is marked on the vertical edge from i to some j̄; (a,e), if the sign (a,e) is marked on the vertical edge from i to some j̄; (e,b), if the sign (e,b) is marked on the vertical edge from i to some j̄; (a,b), if the sign (a,b) is marked on the vertical edge from i to some j̄. The set of elements −→−→µ that arise in this way is denoted by vfr,s, and the same notation is used to represent the set of elements −→−→ν that arise. This is considered to as the set of partial one-row (r,s,f) diagrams. Lemma 5.1. An involution i from −→−→ Dr,s(l) to −→−→ Dr,s(l) is defined by transferring a diagram −→−→ d ∈ −→−→ Dr,s(l) to a diagram i( −→−→ d ) ∈ −→−→ Dr,s(l), which is the reflection in the horizontal axis of that diagram −→−→ d . Proof. It is obvious that i2 = i since the diagram −→−→ d ∈ −→−→ Dr,s(l) will be the same if we reflect it two times around its horizontal axis. We now prove the algebra Vm ⊗Vm ⊗K((Z2 ×Z2) oSr−m × (Z2 ×Z2) oSs−m) is an inflation of the group algebra of the group (Z2 ×Z2) oSr−m ×(Z2 ×Z2) oSs−m along a free K-module Vm of rank |vmr,s| with respect to some bilinear form φm from Vm⊗Vm → K((Z2×Z2)oSr−m×(Z2×Z2)oSs−m) which we will define in the following lemma. Also the multiplication is defined on the basis elements of the algebra Vm⊗Vm⊗K((Z2 ×Z2) oSr−m×(Z2 ×Z2) oSs−m) as follows. For −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 , −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ∈ Vm ⊗Vm ⊗K((Z2 ×Z2) oSr−m × (Z2 ×Z2) oSs−m) we have, ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 ).( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) = −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1φm( −→−→ν 1, −→−→µ 2)ρ(µν)2. (3) Lemma 5.2. Fix an index m, then the K-algebra −→−→ Am/ −→−→ Am+1 is isomorphic to the algebra Vm ⊗Vm ⊗ K((Z2 ×Z2) oSr−m × (Z2 ×Z2) oSs−m) with respect to some bilinear form which we will describe later in the proof. Proof. By applying the partial one row diagram representation of a walled Klein-4 Brauer diagram that we mentioned previously we can define a K-isomorphism Ψ : Vm⊗Vm⊗K((Z2×Z2)oSr−m×(Z2× Z2) oSs−m) → −→−→ Am/ −→−→ Am+1 as follows, 12 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Ψ( −→−→µ ⊗ −→−→ν ⊗ρ(µν)) = Y−→−→µ,−→−→ν ,ρ(µν) We now define the bilinear form φm, which is required for multiplication of elements from the algebra Vm ⊗ Vm ⊗ K((Z2 × Z2) o Sr−m × (Z2 × Z2) o Ss−m). For −→−→µ 1 ⊗ −→−→ν 1 ⊗ ρ(µν)1, −→−→µ 2 ⊗ −→−→ν 2 ⊗ ρ(µν)2 ∈ Vm ⊗Vm ⊗K((Z2 ×Z2) oSr−m × (Z2 ×Z2) oSs−m) we have, Ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1) = Y−→−→µ 1,−→−→ν 1,ρ(µν)1 ∈ −→−→ Dr,s(l) and Ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2) = Y−→−→µ 2,−→−→ν 2,ρ(µν)2 ∈ −→−→ Dr,s(l) Using the multiplication of the elements of the walled klein-4 diagram algebra −→−→ Dr,s(l), that we described in the section 3.1 we have, Y−→−→µ 1, −→−→ν 1,ρ(µν)1 .Y−→−→µ 2, −→−→ν 2,ρ(µν)2 = lt. −−→−−→ (µν) (4) where t is the number of closed loops in the product (4), and −−→−−→ (µν) ∈ −→−→ Dr,s(l) having 2m or more signed horizontal edges. If the product (4), does not have vertical pair −−−−−−−−−−→−−−−−−−−−−→ (r −m,s−m) then, set φm( −→−→ν 1, −→−→µ 2) = 0, otherwise φm( −→−→ν 1, −→−→µ 2) = lt.ρµν; where ρ(µν) ∈ K((Z2 ×Z2) oSr−m × (Z2 ×Z2) oSs−m) such that, Y−→−→µ 1, −→−→ν 1,ρ(µν)1 ·Y−→−→µ 2,−→−→ν 2,ρ(µν)2 = lt · −−→−−→ (µν) = lt ·Y−→−→µ 1,−→−→ν 2,ρ(µν)1ρ(µν)ρ(µν)2 So we have, Ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 · −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) = Ψ( −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1φm( −→−→µ 1, −→−→ν 2)ρ(µν)2 ) = Ψ( −→−→µ 1 ⊗ −→−→ν 2 ⊗ρ(µν)1 (ltρ(µν))ρ(µν)2 ) = lt ·Y−→−→µ +1 ⊗ −→−→ν −1 ⊗ρ(µν)1ρ(µν)ρ(µν)2 = Y−→−→µ 1, −→−→ν 1,ρ(µν)1 ·Y−→−→µ 2,−→−→ν −2 ,ρ(µν)2 = Ψ( −→−→µ 1 ⊗ −→−→ν −1 ⊗ρ(µν)1 ) · Ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) Ψ is an algebra homomorphism. Suppose that, Ψ( −→−→µ 1 ⊗ −→−→ν 1 ⊗ρ(µν)1 ) = Ψ( −→−→µ 2 ⊗ −→−→ν 2 ⊗ρ(µν)2 ) then, Y−→−→µ 1, −→−→ν 1,ρ(µν)1 = Y−→−→µ +2 , −→−→ν 2,ρ(µν)2 So we have −→−→µ 1 = −→−→µ 2, −→−→ν 1 = −→−→ν 2, ρ(µν)1 = ρ(µν)2. Now let −−→−−→ (µν) ∈ Am/Am+1 then −−→−−→ (µν)= Y−→−→µ +, −→−→ν −,ρ(µν) and Ψ( −→−→µ ⊗ −→−→ν ⊗ ρ(µν)) = Y−→−→µ,−→−→ν ,ρ(µν) = −−→−−→ (µν), here −→−→µ ⊗ −→−→ν ⊗ρ(µν) ∈ Vm⊗Vm ⊗ K((Z2×Z2)oSr−m×(Z2×Z2)oSs−m). Hence Ψ is an isomorphism. Lemma 5.3. The involution i from −→−→ Am/ −→−→ Am+1 to −→−→ Am/ −→−→ Am+1 under the map Ψ corresponds to the standard involution on Vm ⊗Vm ⊗ K((Z2 ×Z2) oSr−m ×(Z2 ×Z2) oSs−m), which sends −→−→µ ⊗ −→−→ν ⊗ρ(µν) to −→−→ν ⊗ −→−→µ ⊗ρ(µν)−. 13 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 Proof. The involution on −→−→ Am/ −→−→ Am+1 is derived from the involution on −→−→ Dr,s(l) stated in lemma 5.1. As a result, the proof follows from the previous lemma 5.2. Lemma 5.4. Let −→−→ d 1 ∈ −→−→ Am/ −→−→ Am+1 and −→−→ d 2 ∈ −→−→ An/ −→−→ An+1 be two walled Klein-4 Brauer diagrams from−→−→ Dr,s(l) and let their respective Ψ-pre images be −→−→µ 1 ⊗ −→−→ν 1 ⊗ ρ(µν)1 and −→−→µ 2 ⊗ −→−→ν 2 ⊗ ρ(µν)2. We assume that n ≥ m. Then the product −→−→ d 1. −→−→ d 2 is either an element of −→−→ An/ −→−→ An+1, or is an element of −→−→ An+1. Under Ψ it corresponds to the scalar multiple of an element −→−→ µ′ ⊗ −→−→µ 2 ⊗ηρ(µν)2; where −→−→ µ′ ∈ Vn and η ∈ K((Z2 ×Z2) oSr−n × (Z2 ×Z2) oSs−n). In the case of n ≤ m, a similar assumption holds true. Proof. Both statements have a proof that is extremely similar to lemma 5.2. As a result of the lemma 5.4, we can now define the bilinear mappings α,β,γ described in [5], which are required for iterated inflation. We have proven the following theorem in total. Theorem 5.5. The walled Klein-4 Brauer algebra −→−→ Dr,s(l) is an iterated inflation of group algebras of groups ((Z2 ×Z2) oSr−m ×(Z2 ×Z2) oSs−m), where 0 ≤ m ≤ min(r,s). Specifically as a free K-module−→−→ Dr,s is equal to K((Z2 ×Z2) oSr × (Z2 ×Z2) oSs) ⊕ (V1 ⊗V1 ⊗K((Z2 ×Z2) oSr−1 × (Z2 ×Z2) oSs−1)) ⊕ (V2 ⊗V2 ⊗ K((Z2×Z2)oSr−2×(Z2×Z2)oSs−2)) ⊕ ··· , and inflation begins with K((Z2×Z2)oSr×(Z2×Z2)oSs) inflates it along V1 ⊗V1 ⊗K((Z2 ×Z2) oSr−1 ×(Z2 ×Z2) oSs−1) and so on, concluding with an inflation of K((Z2 ×Z2) oS1) or K((Z2 ×Z2) oS0) as bottom layer depending on whether (r + s) is even or odd. Proof. The proof follows from the above lemma 5.2, lemma 5.3 and lemma 5.4. Corollary 5.6. The walled Klein-4 Brauer algebra −→−→ Dr,s(l) over any field of characteristic p = 0 or p > max(r,s) is cellular. Proof. From the theorem 5.5, the walled Klein-4 Brauer algebra −→−→ Dr,s is an iterated inflation of group algebras of the group ((Z2 ×Z2) o Sr−m × (Z2 ×Z2) o Ss−m) for 0 ≤ m ≤ min(r,s) along Vm. So the proof follows from the remarks 2.5 and 2.6. Corollary 5.7. If l 6= 0 or r 6= s then the simple modules of the walled Klein-4 Brauer algebras −→−→ Dr,s(l) are indexed by all pair (m,λL,λR), where 0 ≤ m ≤ min(r,s) and (λL,λR) ∈ ∧r−m,s−m4−reg . If l = 0 and r = s we get the same indexing set for the simple modules as above but with the single simple corresponding to l = min(r,s). Proof. By theorem 4.5, for l 6= 0 or r 6= s, the simple modules of the walled Klein-4 Brauer algebras−→−→ Dr,s(l) are indexed by all pair (m,λL,λR), where 0 ≤ m ≤ min(r,s) and (λL,λR) ∈ ∧r−m,s−m4−reg . In the case of l = 0, the above assertion is also valid except that the case m = 0 (which occurs only for r even) does not contribute a simple module [6]. The exception in this situation arises from the fact that in the iterated inflation, there is a piece with φm equal to zero [6]. Here φm is the bilinear form defined in the lemma 5.2. 14 AR TI CL E IN PR ES S A. Tamilselvi, S. Dhilshath / J. Algebra Comb. Discrete Appl. -(-) (2023) 1–15 References [1] A. G. Cox, P. P. Martin, A. E. Parker, C. Xi, Representation theory of towers of recollement: Theory, notes, and examples, J. Algebra 302 (2006) 340–360. [2] C. W. Curtis, I. Reiner, Methods of representation theory, vol. 1, Wiley (1981). [3] J. A. Green, Polynomial representations of GLn, Lecture notes in mathematics 830, Springer, Berlin (1980). [4] B. Kethesan, The Structure of walled signed Brauer algebras, Kyungpook Math. J 56 (2016) 1047– 1067. [5] S. Konig, C. Xi, Cellular Algebras: Inflations and Morita equivalences, J. London Math. Soc. 60(3) (1999) 700–722. [6] S. Konig, C. Xi, A characteristic free approach to Brauer algebras, Trans. Am. Math. Soc. 353(4) (2001) 1489–1505. [7] M. Parvathi, D. Savithri, Representation of G-Brauer algebras, Southeast Asian Bulletin of Mathe- matics 26(3) (2002) 453–468. [8] M. Parvathi, B. Sivakumar, The Klein-4 diagram algebras, J. Algebra Appl. 07(02) (2008) 231–262. [9] B. Sivakumar, Matrix units for the group algebra kGf = k((Z2 ×Z2) oSf ), Asian-Eur. J. Math. 2(2) (2009) 255–277. [10] A. Tamilselvi, Cellularity and representations of walled cyclic G-Brauer algebras, Malaya J. Mat. 9(1) (2021) 402–408. [11] A. Tamilselvi, A. Vidhya, B. Kethesan, Robinson-Schensted Correspondence for the walled Brauer algebras and the walled signed Brauer algebras, Algebra and its applications, Springer Proc. Math. Stat., Springer, Singapore, 174 (2016) 195-223. 15 https://doi.org/10.1016/j.jalgebra.2006.01.009 https://doi.org/10.1016/j.jalgebra.2006.01.009 http://dx.doi.org/10.5666/KMJ.2016.56.4.1047 http://dx.doi.org/10.5666/KMJ.2016.56.4.1047 https://doi.org/10.1112/S0024610799008212 https://doi.org/10.1112/S0024610799008212 https://www.ams.org/journals/tran/2001-353-04/S0002-9947-00-02724-0/S0002-9947-00-02724-0.pdf https://www.ams.org/journals/tran/2001-353-04/S0002-9947-00-02724-0/S0002-9947-00-02724-0.pdf https://doi.org/10.1007/s10012-002-0453-6 https://doi.org/10.1007/s10012-002-0453-6 https://doi.org/10.1142/S0219498808002795 https://doi.org/10.1142/S1793557109000212 https://doi.org/10.1142/S1793557109000212 https://doi.org/10.26637/MJM0804/0095 https://doi.org/10.26637/MJM0804/0095 https://doi.org/10.1007/978-981-10-1651-6_11 https://doi.org/10.1007/978-981-10-1651-6_11 https://doi.org/10.1007/978-981-10-1651-6_11 Introduction Preliminaries Walled Klein-4 Brauer algebras Indexing set for the simple modules of the walled Klein-4 Brauer algebras Walled Klein-4 Brauer algebras are iterated inflations References