The invariant theory of classical groups, algebraic Lie theory, algebraic number theory, knot theory, integrable models and statistical mechanics, quantum computing are the few areas of diagram algebras arising in different areas of mathematics and physics. In order to characterise invariants of classical groups acting on tensor powers of the vector representations, Brauer [2] introduced a new class of algebras called Brauer algebras. The Brauer algebras used graphs to represent its basis. Hence it can be considered as a class of diagram algebras, that are finite dimensional algebras whose basis consists of diagrams. These basis have interesting combinatorial properties to be studied in their own right.

Parvathi and Kamaraj [10] introduced a new class of algebras called signed Brauer algebras $S_f^{(x)}$ which are a generalization of Brauer algebras. Parvathi and Selvaraj [12] studied signed Brauer algebras as a class of centraliser algebras, which are the direct product of orthogonal groups over the field of real numbers $ℝ$. Parvathi and Savithri [11] introduced a new class of algebras called G-Brauer algebras $D_f^G\left(x\right)$, where G is abelian, which are a generalization of signed Brauer algebras $S_f^{(x)}$ introduced by [10] and Brauer algebras.

Brown [3, 4], Hanlon and Wales [6, 7] and Wenzl [14] studied the Brauer algebras by using diagrams to represent its basis and Young diagrams to represent its irreducible representations. Brown has not discussed the structure of Brauer algebras when the radical is non zero. This study was carried out by Hanlon and Wales in [6]. They determined the structure of the radical of a non-semisimple Brauer algebras by introducing the notion of 1-factor, m,k-partial 1-factor and the combinatorially defined matrix $T_{m,k}^{\lambda }(x)$. In [7], they used these matrices to find the eigen values and eigen vectors corresponding to Brauer's centraliser algebras.

However for signed Brauer algebras, the eigen values corresponding to a non-semisimple signed Brauer algebras have not been dealt completely. This motivated us to study the eigen values for the signed Brauer algebras [13] and G-Brauer algebras where $G={\mathbb{Z}}_r$.

In this paper, we analyse the irreducible representations of $D_f^G\left(x\right)$ by determining the quotient ${\overrightarrow{I}}_f^G(x,2k)$ of $D_f^G\left(x\right)$ by its radical. We also find the eigenvalues and eigenspaces of certain symmetric matrices ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$ for some values of m and k using the representation theory of the generalised symmetric group. The matrices $T_{m,k}^{[\lambda ]}(x)$ whose entries are indexed by generalised m,k signed partial 1-factors, which helps in determining the non semisimplicity of these cyclic G-Brauer algebras $D_f^G\left(x\right)$, where $G={\mathbb{Z}}_r$. In this paper G refers to ${\mathbb{Z}}_r,r>0$.

We begin by recalling some known results in the representation theory of the generalised symmetric group [1, 5, 8].

Definition 2.1. For each standard multi-tableaux [s] and [t] of shape [λ],

$$m_{[s][t]}=\pi {\displaystyle \sum _{\sigma \in R_{[t]}}\sigma },\pi \in D([t])$$

where $\pi [s]=[t]$, $D([t])$ is the left coset representative of $R_{[t]}$, row stabiliser of [t].

Definition 2.2. For each multi-partition $[\lambda ]$, there exists two sided ideals $K^{[\lambda ]}$ and $\overline{K^{[\lambda ]}}$ of ${\widehat{\widehat{S}}}_m$, group algebras of the generalised symmetric group.

• 1. $K^{[\lambda ]}=$ Linear span $\{m_{[s][t]}/[s]$ and [t] are standard multi-tableaux of shape $[\mu ]$ for $[\mu ]⊵[\lambda ]\}$.

• 2. $\overline{K^{[\lambda ]}}=$ Linear span $\{m_{[s][t]}/[s]$ and [t] are standard multi-tableaux of shape $[\mu ]$ for $[\mu ]⊳[\lambda ]\}$.

• 3. ${\widehat{\widehat{S}}}_m^{[\lambda ]}=$ Linear span $\{m_{[s][t]}/[s]$ and [t] are standard multi-tableaux of shape $[\lambda ]\}$.

Remark 2.3. • For each multi-partition $[\lambda ]$ of m, let ${\overrightarrow{S}}^{[\lambda ]}$ denote the Specht module corresponding to $[\lambda ]$ and let $d_{[\lambda ]}$ denote the dimension of ${\overrightarrow{S}}^{[\lambda ]}$.

• The ideal ${\widehat{\widehat{S}}}_m^{[\lambda ]}$ considered as a vector space of linear transformations of ${\overrightarrow{S}}^{[\lambda ]}$ is the full matrix algebras $\text{End}({\overrightarrow{S}}^{[\lambda ]})$.

Definition 2.4. There exists left ideals ${\overrightarrow{I}}_1,{\overrightarrow{I}}_2,\dots ,{\overrightarrow{I}}_{d_{[\lambda ]}}$, right ideals ${\overrightarrow{J}}_1,{\overrightarrow{J}}_2$, $\dots ,{\overrightarrow{J}}_{d_{[\lambda ]}}$ and the unique minimal two sided ideal ${\widehat{\widehat{S}}}_m^{[\lambda ]}$ of ${\widehat{\widehat{S}}}_m$, group algebras of the generalised symmetric group. ${\widehat{\widehat{S}}}_m^{[\lambda ]}$ can be written either as direct sum of simple left ideals

$${\widehat{\widehat{S}}}_m^{[\lambda ]}={\overrightarrow{I}}_1\oplus \cdots \oplus {\overrightarrow{I}}_{d_{[\lambda ]}},$$

or as a direct sum of simple right ideals

$${\widehat{\widehat{S}}}_m^{[\lambda ]}={\overrightarrow{J}}_1\oplus \cdots \oplus {\overrightarrow{J}}_{d_{[\lambda ]}}$$

where each ${\overrightarrow{I}}_i$ is a left ideal of ${\widehat{\widehat{S}}}_m$ for which multiplication on the left gives a representation isomorphic to ${\overrightarrow{S}}^{[\lambda ]}$ and each ${\overrightarrow{J}}_i$ is a right ideal of ${\widehat{\widehat{S}}}_m$ for which right multiplication is isomorphic to ${\overrightarrow{S}}^{[\lambda ]}$.

There exists a basis $A_1,\dots ,A_{d_{[\lambda ]}}$ for ${\overrightarrow{S}}^{[\lambda ]}$ with respect to which the matrices ${\Psi }_{[\lambda ]}(\sigma )$ for $\sigma \in {\widehat{\widehat{S}}}_m$ acting on ${\overrightarrow{S}}^{[\lambda ]}$ are orthogonal. i.e. ${\Psi }_{[\lambda ]}({\sigma }^{-1})={\Psi }_{[\lambda ]}{(\sigma )}^t$. For any elements $x_i,y_j\in {\widehat{\widehat{S}}}_m^{[\lambda ]},1\le i,j\le d_{[\lambda ]}$, choose $x_i,y_j$ so that

(1.1)$$x_i{y}_j{x}_r{y}_l=\left\{\begin{array}{ll}{x}_i{y}_l,\hfill & ifj=r;\hfill \\ 0,\hfill & \text{otherwise}\text{.}\hfill \end{array}\right.$$

Definition 3.1. A r-signed 1-factor on 2f vertices is a signed diagram with f vertices arranged in two rows and f, r-signed edges such that each vertex is incident to exactly one r-signed edge, labeled by the primitive $r^{th}$-root of unity ${\xi }^1,{\xi }^2,\dots ,{\xi }^r$. A r-signed edge is said to be an i-edge if it is labeled by ${\xi }^i$. The set of all r-signed 1-factor on 2f vertices is denoted by $P_f^G$, where G is assumed to be a cyclic group.

A r-signed 1-factor $\delta \in P_f^G$ will be represented as a diagram having two rows of f vertices each, the f vertices in the top row are labeled by $1,2,\dots ,f$ from left to right and the f vertices in the bottom row are labeled by $f+1,f+2,\dots ,2f$ from left to right. There are $r^f(2f-1)!!=r^f\cdot 1\cdot 3\cdots (2f-1)$ ways of joining these 2f vertices which is incident to exactly to one r-signed edge.

Example 3.2. The 27 r-signed 1-factors in $P_2^{{\mathbb{Z}}_3}$ is as follows.

An r-signed edge of $\delta \in P_f^G$ is called r-signed horizontal edge if it joins two vertices in the same row of $\delta \in P_f^G$.

An r-signed edge of $\delta \in P_f^G$ is called r-signed vertical edge if it joins two vertices in different rows of $\delta \in P_f^G$.

Definition 3.3. Let ${\overrightarrow{V}}_f$ be the vector space over a field K with $P_f^G$ as its basis.

Definition 3.4. Let ${\overrightarrow{V}}_f(2k)=$ Linear span$\{\delta \in P_f^G/$the number of r-signed horizontal edges in $\delta \ge 2k\}$, $k=0,1,\dots ,⌊\frac{f}{2}⌋$ which is a subspace of ${\overrightarrow{V}}_f$. ${\overrightarrow{V}}_f(2k)$ can be written as direct sum of vector subspaces ${\overrightarrow{V}}_f^{*}(2m),k\le m\le ⌊\frac{f}{2}⌋$ spanned by all r-signed 1-factors having exactly 2m r-signed horizontal edges.

$$\overrightarrow{V_f}(2k)={\overrightarrow{V_f}}^{*}(2k)\oplus {\overrightarrow{V_f}}^{*}(2k+2)\oplus \dots \oplus {\overrightarrow{V_f}}^{*}\left(2⌊\frac{f}{2}⌋\right).$$

In the following, we make ${\overrightarrow{V}}_f$ as an algebra over the field K(x), where K is any field and x is an indeterminate, by defining composition of two elements ${\delta }_1,{\delta }_2\in P_f^G$.

For ${\delta }_1,{\delta }_2\in P_f^G$, the graph ${\overrightarrow{U}}_{{\delta }_2}^{{\delta }_1}$ with 3f vertices arranged in three rows with the first row, the top row of ${\delta }_1$, the second row is obtained by identifying the vertices in the bottom row of ${\delta }_1$ with the vertices in the top row of ${\delta }_2$ and the third row, the bottom row of ${\delta }_2$. The graph ${\overrightarrow{U}}_{{\delta }_2}^{{\delta }_1}$ consists of exactly f, r-signed paths ${\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_f$, some number $m_i({\delta }_1,{\delta }_2))$ of i-cycles for $i=1,2,\dots ,r$ such that

• 1. The r-signed path ${\mathcal{P}}_i$ contains one or more r-signed edges. The initial and endpoint of ${\mathcal{P}}_i$ does not meet each other.

• 2. Each i-cycle is of even length consisting some number k_j of i_j-edges, $1\le i_j\le r$ with ${\displaystyle \sum k_j}{i}_j\equiv i(\mathrm{mod}r)$, entirely of vertices lying in the middle row.

Example 3.5. For ${\delta }_1,{\delta }_2\in P_5^{{\mathbb{Z}}_3}$, the diagram in ${\overrightarrow{U}}_{{\delta }_2}^{{\delta }_1}$ is

Definition 3.6. Let ${\delta }_1$ and ${\delta }_2$ be r-signed 1-factors in $P_f^G$. Define the composition of r-signed diagrams ${\delta }_1\circ {\delta }_2$ to be the r-signed 1-factor in the following way

• 1. The top (respectively bottom) row of ${\delta }_1\circ {\delta }_2$ have the same r-signed horizontal edges in the top (respectively bottom) row of ${\delta }_1$ (respectively ${\delta }_2)$.

• 2. The vertices u and v are adjacent if and only if there is a r-signed path ${\mathcal{P}}_i$ in ${\overrightarrow{U}}_{{\delta }_2}^{{\delta }_1}$ joining u to v and an edge joining u to v is an i-edge if the path contains some number k_j of i_j-edges, $1\le i_j\le r$ with ${\displaystyle \sum k_j}{i}_j\equiv i(\mathrm{mod}r)$.

Definition 3.7. The cyclic G-Brauer algebra $D_f^G\left(x\right)$ is an associative algebras over the field K[x] with basis $P_f^G$ and the multiplication * of r-signed 1-factors given by

$${\delta }_1*{\delta }_2=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i({\delta }_1,{\delta }_2)}({\delta }_1\circ {\delta }_2),\text{ where }G={\mathbb{Z}}_r$$

This algebras $D_f^G\left(x\right)$ is called the G-Brauer algebras defined in [11], when $G={\mathbb{Z}}_r$.

Example 3.8. For ${\delta }_1,{\delta }_2\in P_5^{{\mathbb{Z}}_3}$, the diagram in ${\delta }_1*{\delta }_2$ is

Let $I_f^G(x,2k)=$Linear span$\{\delta \in P_f^G/$number of r-signed horizontal edges in $\delta \ge 2k\}$. Clearly by the multiplication defined above $I_f^G(x,2k)$ is an ideal of $D_f^G\left(x\right)$.

Let ${\overrightarrow{I}}_f^G(x,2k)=$Linear span$\{\delta \in P_f^G/$number of r-signed horizontal edges in δ is equal to $2k\}$. ${\overrightarrow{I}}_f^G(x,2k)$ denotes the quotient $I_f^G(x,2k)/I_f^G(x,2k+2)$.

To describe the structure of the quotients ${\overrightarrow{I}}_f^G(x,2k)$ in terms of the eigenvalues and eigenspaces of certain matrices.

Definition 4.1. A generalised m,k signed partial 1-factor on f=m+2k vertices is a r-signed diagram whose vertices are arranged in a single row with k, r-signed horizontal edges and m free vertices.

Let $P_{m,k}^G$ denotes the set of all generalised m,k signed partial 1-factors and let $V_{m,k}^G$ be the real vector space with basis $P_{m,k}^G$.

The generalised symmetric group on m symbols $\{1,2,\dots ,m\}$ is denoted by $S_m^G$.

Let us now define $\pi \in S_m^G$ by $\pi (i)=(\tau (i),\sigma (i))$ where $\sigma \in S_m$ and the function $\tau :\underset{\_}{m}\to \underset{\_}{r}$, where $\underset{\_}{m}$ denotes the set $\text{{}1,2,\dots ,m\}$ and $\underset{\_}{r}$ denotes the set $\text{{}\xi ,{\xi }^2,\dots ,{\xi }^r\}$, ${\xi }^i$'s are primitive r^th root of unity.

Definition 4.2. Let f₁ (respectively f₂) be generalised m,k signed partial 1-factors with the free vertices of f₁ (respectively f₂) is labeled by ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_m$ (respectively ${\beta }_1<{\beta }_2<\cdots <{\beta }_m$).

The union of f₁ and f₂ is a r-signed graph obtained by identifying i-th vertex of f₁ with the i-th vertex of f₂ consists some number $m_i(f_1,f_2)$ of disjoint i-cycles together with m disjoint r-signed paths ${\mathcal{P}}_1,\dots ,{\mathcal{P}}_m$ whose endpoints are in the set $\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m,{\beta }_1,{\beta }_2,\dots ,{\beta }_m\}$.

Definition 4.3. Let $f_1,f_2\in P_{m,k}^G$. Define an inner product $<f_1,f_2>$ on $V_{m,k}^G$ as follows.

• 1. If any r-signed path ${\mathcal{P}}_i$ joins a α_j to a α_i (or equivalently a β_j to a β_i) then $<f_1,f_2>=0$.

• 2. If each r-signed path ${\mathcal{P}}_i$ joins β_i to ${\alpha }_{\sigma i}$ and some number k_j of i_j edges $1\le i_j\le r$ with ${\displaystyle \sum _{i_j=1}^r{k}_j}{i}_j\equiv \tau (i)(\text{mod }r)$, then

$$<f_1,f_2>=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(f_1,f_2)}\pi ,\text{ where }\pi =(\tau ,\sigma )\in S_m^G,\sigma \in S_m.$$

Note. $<f_1,f_2>=<f_2^{✠},f_1^{✠}>$, where ✠ is the anti-isomorphism defined on the algebras $K{S}_m^G$ by $\sigma \to {\sigma }^{-1},\sigma \in S_m^G$.

Proposition 4.4. Let f=m+2k. Then the quotient ${\overrightarrow{I}}_f^G(x,2k)$ is isomorphic as algebras to $(V_{m,k}^G\otimes V_{m,k}^G\otimes K{S}_m^G,\cdot )$, where

$$(a\otimes b\otimes {\pi }_1)\cdot (c\otimes d\otimes {\pi }_2)=a\otimes d\otimes ({\pi }_1<b,c>{\pi }_2),a,b,c,d\in P_{m,k}^G,{\pi }_1,{\pi }_2\in S_m^G.$$

Proof. The proof follows as in [6]. Instead of m,k partial 1-factors and symmetric group, the generalised m,k signed partial 1-factors and the generalised symmetric group are used to prove the theorem, we give it here for the sake of completion.

As a vector space ${\overrightarrow{I}}_f^G(x,2k)$ has basis the set of all r-signed 1-factors with exactly 2k r-signed horizontal edges.

Consider the linear map $\varphi :V_{m,k}^G\otimes V_{m,k}^G\otimes K{S}_m^G\to {\overrightarrow{I}}_f^G(x,2k)$.

Given $f_1,f_2\in P_{m,k}^G$, we define $\varphi (f_1\otimes f_2\otimes \sigma )$ to be the r-signed 1-factor on 2f vertices in the following way.

Let f₁ be the generalised m,k signed partial 1-factor with free vertices ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_m$ and let f₂ be the generalised m,k signed partial 1-factor with free vertices ${\beta }_1<{\beta }_2<\cdots <{\beta }_m$ and given $\sigma \in S_m^G$ such that

• 1. A r-signed horizontal edge joining i to j in the top row if and only if i and j are joined by an r-signed horizontal edge in f₁.

• 2. A r-signed horizontal edge joining f+i to f+j in the bottom row if and only if i and j are joined by an r-signed horizontal edge in f₂.

• 3. A r-signed vertical edge joining α_i to ${\beta }_{\sigma (i)}$ for $i=1,2,\cdots m$.

The linear map ϕ defined in this way is clearly 1-1 and onto. Hence it is a vector space isomorphism of $V_{m,k}^G\otimes V_{m,k}^G\otimes K{S}_m^G$ onto ${\overrightarrow{I}}_f^G(x,2k)$.

It remains to show that ϕ is multiplicative.

$$\text{ i}\text{.e }\varphi (d_1\cdot d_2)=\varphi (d_1)\circ \varphi (d_2),d_1,d_2\in V_{m,k}^G\otimes V_{m,k}^G\otimes K{S}_m^G.$$

Let us now assume that $d_1=a\otimes b\otimes {\pi }_1$ and $d_2=c\otimes d\otimes {\pi }_2$ where a,b,c,d be the generalised m,k signed partial 1-factors with free vertices ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_m$, ${\beta }_1<{\beta }_2<\cdots <{\beta }_m$ , ${\gamma }_1<{\gamma }_2<\cdots <{\gamma }_m$, ${\psi }_1<{\psi }_2<\cdots <{\psi }_m$ respectively and ${\pi }_1,{\pi }_2\in S_m^G$.

Consider the product in ${\overrightarrow{I}}_f^G(x,2k)$

$$\varphi (d_1)\circ \varphi (d_2)=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(d_1,d_2)}{d}_3,d_3\in D_f^G\left(x\right).$$

Case 1. Suppose there is a r-signed path joining α_i to α_j or ψ_i to ψ_j in ${\overrightarrow{U}}_{\varphi (d_2)}^{\varphi (d_1)}$ then $\varphi (a\otimes b\otimes {\pi }_1)\circ \varphi (c\otimes d\otimes {\pi }_2)=0=\varphi ((a\otimes b\otimes {\pi }_1)\cdot (c\otimes d\otimes {\pi }_2))$. Therefore

$$\varphi (d_1\cdot d_2)=\varphi (d_1)\circ \varphi (d_2).$$

Case 2. Suppose there is a r-signed path joining α_i to ${\psi }_{\sigma (i)}$ in ${\overrightarrow{U}}_{\varphi (d_2)}^{\varphi (d_1)}$ for $i=1,2,\dots ,m$, then $\varphi (d_1)\circ \varphi (d_2)=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i({\delta }_1,{\delta }_2)}{d}_3=\varphi (d_1\cdot d_2)$.

Hence ϕ is an algebra isomorphism.

To describe the structure of the ring ${\overrightarrow{I}}_f^G(x,2k)$ in terms of the eigenvalues of certain matrices.

Definition 5.1. Let ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$ be the $(p{d}_{[\lambda ]})$-by-$(p{d}_{[\lambda ]})$ matrix which is $d_{[\lambda ]}$-by-$d_{[\lambda ]}$ blocks of p-by-p matrices where p is the number of generalised m,k signed partial 1-factors. The matrices in the each block are indexed by pairs of generalised m, k signed partial 1-factors being ${\psi }_{[\lambda ]}(<b,c>)$, for the corresponding r-signed 1-factor.

Let ${\overrightarrow{N}}^{[\lambda ]}$ denotes the null space of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$, the matrix corresponding to generalised m,k signed partial 1-factors and the multi partition [λ] and ${\overrightarrow{R}}^{[\lambda ]}$ denotes the range of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$, the matrix corresponding to generalised m,k signed partial 1-factors and the multi partition [λ].

Note. If $<b,c>=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i)(b,c)}\sigma$ then $<c,b>=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(c,b)}{\sigma }^{-1}$. So the matrix ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$ is symmetric.

Choose a basis $u^{(1)},\dots ,u^{(n)}$ for ${\overrightarrow{N}}^{[\lambda ]}$ and an orthonormal basis of eigenvectors $v^{(1)},\dots ,v^{(r)}$ for the nonzero eigenvalues ${\mu }^{(1)},\dots ,{\mu }^{(r)}$.

Definition 5.2. For given any left ideal ${\overrightarrow{I}}_t$ and a generalised m, k signed partial 1-factor d, define

$${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)=\text{ Linear Span }\{c\otimes d\otimes x/c\in P_{m,k}^G,x\in {\overrightarrow{I}}_t\}.$$

Lemma 5.3. ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ is a left ideal of ${\overrightarrow{I}}_f^G(x,2k)$.

Proof. The proof follows as in [6]. Instead of m,k partial 1-factors and symmetric group, the generalised m,k signed partial 1-factors and the generalised symmetric group are used to prove the theorem, we give it here for the sake of completion.

For $c\otimes d\otimes x\in {\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$, choose $\delta \in {\overrightarrow{I}}_f^G(x,2k)$ such that $\delta *(c\otimes d\otimes x)$ not equal to zero, that is $\delta *(c\otimes d\otimes x)$ and $c\otimes d\otimes x$ have same number of r-signed horizontal edges.

$$\begin{array}{l}(a\otimes b\otimes \pi )*(c\otimes d\otimes x)=(a\otimes d\otimes \pi <b,c>x),\pi \in S_m^G,x\in {\overrightarrow{I}}_t\\ =(a\otimes d\otimes \pi x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(b,c)}{\pi }_{1}x),\\ \text{ where }\pi ,{\pi }_1\in S_m^G,x\in {\overrightarrow{I}}_t\\ =x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(b,c)}(a\otimes d\otimes \pi {\pi }_{1}x),\\ (\pi {\pi }_1\in S_m^G\text{ by the definition of }{S}_m^G)\end{array}$$

Since $x\in {\overrightarrow{I}}_t$ and ${\overrightarrow{I}}_t$ is a left ideal of ${\widehat{\widehat{S}}}_m$, $y=\pi {\pi }_{1}x\in {\overrightarrow{I}}_t$. Hence

$$(a\otimes b\otimes \pi )*(c\otimes d\otimes x)=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(b,c)}(a\otimes d\otimes y)\in {\overrightarrow{V}}_L({\overrightarrow{I}}_t,d).$$

Therefore ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ is a left ideal of ${\overrightarrow{I}}_f^G(x,2k)$.

Define ${\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)\subset {\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ to be the linear span of all

$$\sum {(u)}_{(c,i)}\left(c\otimes d\otimes {\overrightarrow{A}}_{i,t}\right),$$

where u is in ${\overrightarrow{N}}^{[\lambda ]}$ and ${\overrightarrow{A}}_{i,t}$ is the basis element of ${\overrightarrow{I}}_t$ corresponding to the basis element ${\overrightarrow{A}}_i$ in ${\overrightarrow{S}}^{[\lambda ]}$. i.e. the linear span of the set of all elements mapped to zero in ${\overrightarrow{I}}_f^G(x,2k)$.

Proposition 5.4. Suppose $v={\displaystyle \sum {(v)}_{(c,i)}}\left(c\otimes d\otimes {\overrightarrow{A}}_{i,t}\right)\in {\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$. Let a, b be generalised m,k signed partial 1-factors. For any $\sigma \in S_m^G$, $(a\otimes b\otimes \sigma )v=a\otimes d\otimes \sigma {\displaystyle \sum {\gamma }_j}{\overrightarrow{A}}_{j,t}$, where γ_j is the (b, j) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)(v)$.

Proof. The proof follows as in [6]. Instead of m,k partial 1-factors and symmetric group, the generalised m,k signed partial 1-factors and the generalised symmetric group are used to prove the theorem, we give it here for the sake of completion.

$$\begin{array}{l}(a\otimes b\otimes \sigma )*{\displaystyle \sum v_{(c,i)}}\left(c\otimes d\otimes {\overrightarrow{A}}_{i,t}\right)=a\otimes d\otimes \sigma {\displaystyle \sum v_{(c,i)}}<b,c>{\overrightarrow{A}}_{i,t}\\ =a\otimes d\otimes \sigma {\displaystyle \sum {(v)}_{c,i}}{x}^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(b,c)}{\sigma }_1{A}_{i,t}\\ =a\otimes d\otimes {\displaystyle \sum {\gamma }_j}{\overrightarrow{A}}_{j,t}\end{array}$$

where ${\sigma }_1{A}_{i,t}=A_{j,t}$ and γ_j is the coefficient of ${\overrightarrow{A}}_{j,t}$ in ${\displaystyle \sum v_{(c,i)}}<b,c>{\overrightarrow{A}}_{j,t}$.

By definition of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$, the coefficient of ${\overrightarrow{A}}_{j,t}$ in $<b,c>{\overrightarrow{A}}_{i,t}$ is the (b, j), (c, i) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$. Thus γ_j is the (b,j) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)(v)$.

Proposition 5.5.

• 1. ${\overrightarrow{I_f}}^G(x,2k){\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)=0$.

• 2. $\overrightarrow{I}(v)={\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ for any v in ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ not in ${\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$.

• 3. ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)/{\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$ is irreducible as a left ${\overrightarrow{I}}_f^G(x,2k)$ module.

Proof. The proof follows as in [6]. Instead of m,k partial 1-factors and symmetric group, the generalised m,k signed partial 1-factors and the generalised symmetric group are used to prove the theorem. We give it here for the sake of completion.

Suppose w is a generating element of ${\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$ and γ_j is the (b,j) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)(v)$. By the definition of ${\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$, γ_j are all 0, for any $(a\otimes b\otimes \sigma )$. Hence ${\overrightarrow{I}}_f^G(x,2k){\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)=0$.

Suppose v is in ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$ but not in ${\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$. Choose γ_j, the (b,j) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)(v)$ is not 0. Then $a\otimes b\otimes \sigma (v)$ is not 0 Note that a and σ were arbitrary. Since ${\overrightarrow{I}}_t$ is an irreducible ${\widehat{\widehat{S}}}_m$ module, the images under $\sigma \in {\widehat{\widehat{S}}}_m$ of any nonzero vector in ${\overrightarrow{I}}_t$ generate all of ${\overrightarrow{I}}_t$. Hence vectors of the form $(a\otimes b\otimes \sigma )*{\displaystyle \sum v_{(c,j)}}\left(c\otimes d\otimes {\overrightarrow{A}}_{j,t}\right)$ generate all of ${\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$. Hence $\overrightarrow{I}(v)={\overrightarrow{V}}_L({\overrightarrow{I}}_t,d)$.

The last part of the proof from the above two.

Let ${\overrightarrow{W}}_L^{[\lambda ]}=\oplus {\overrightarrow{W}}_L({\overrightarrow{I}}_t,d)$ and . By Proposition 5.4, ${\overrightarrow{W}}_L^{[\lambda ]}$ is a nilpotent left ideal of ${\overrightarrow{I}}_f^G(x,2k)$.

Recall that ${\widehat{\widehat{S}}}_m^{[\lambda ]}$ can also be written as a direct sum of right ideals ${\overrightarrow{J}}_1,\dots ,{\overrightarrow{J}}_{d_{[\lambda ]}}$.

For given any right ideal ${\overrightarrow{J}}_t$ and a generalised m,k signed partial 1-factor a, define

$${\overrightarrow{V}}_R({\overrightarrow{J}}_t,a)=\text{ Linear Span }\{a\otimes b\otimes x/b\in P_{m,k}^G,x\in {\overrightarrow{J}}_t\}.$$

Lemma 5.6. ${\overrightarrow{V}}_R({\overrightarrow{J}}_t,c)$ is a right ideal of ${\overrightarrow{I}}_f^G(x,2k)$.

Proof. For $c\otimes d\otimes x\in {\overrightarrow{V}}_R({\overrightarrow{J}}_t,c)$, choose $\delta \in {\overrightarrow{I}}_f^G(x,2k)$ such that $(c\otimes d\otimes x)*\delta$ not equal to zero, that is $(c\otimes d\otimes x)*\delta$ have same number of signed horizontal edges as in $c\otimes d\otimes x$.

$$\begin{array}{l}(c\otimes d\otimes x)*(a\otimes b\otimes \pi )=(c\otimes b\otimes x<d,a>\pi ),\pi \in S_m^G,x\in {\overrightarrow{J}}_t\\ =(c\otimes b\otimes x{x}^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(d,a)}{\pi }_1\pi ),\\ \text{ where }\pi ,{\pi }_1\in S_m^G,x\in {\overrightarrow{J}}_t\\ =x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(d,a)}(c\otimes b\otimes x{\pi }_1\pi ),\\ ({\pi }_1\pi \in S_m^G\text{ by the definition of }{S}_m^G)\end{array}$$

Since $x\in {\overrightarrow{J}}_t$ and ${\overrightarrow{J}}_t$ is a right ideal of ${\widehat{\widehat{S}}}_m$, $y=x{\pi }_1\pi \in {\overrightarrow{J}}_t$,

$$(c\otimes d\otimes x)*(a\otimes b\otimes \pi )=x^{{\displaystyle \sum _{i=1}^{r}i}{m}_i(d,a)}(c\otimes b\otimes y)\in {\overrightarrow{V}}_R({\overrightarrow{J}}_t,c).$$

Therefore ${\overrightarrow{V}}_R({\overrightarrow{J}}_t,c)$ is a right ideal of ${\overrightarrow{I}}_f^G(x,2k)$.

Define ${\overrightarrow{W}}_R({\overrightarrow{J}}_t,a)\subset {\overrightarrow{V}}_R({\overrightarrow{J}}_t,a)$ to be the linear span of all

$${\displaystyle \sum {(u)}_{(c,i)}}\left(a\otimes b\otimes {\overrightarrow{A}}_{j,t}\right)$$

where $u^t{\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)=0$ and ${\overrightarrow{A}}_{j,t}$ is as before.

The same proofs used in Propositions 5.4 and 5.5 shows that

• 1. ${\overrightarrow{W}}_R({\overrightarrow{J}}_t,a)\circ (c\otimes d\otimes \sigma )=0$,

• 2. ${\overrightarrow{V}}_R({\overrightarrow{J}}_t,a)/{\overrightarrow{W}}_R({\overrightarrow{J}}_t,a)$ is an irreducible right ${\overrightarrow{I}}_f^G(x,2k)$ module.

Define ${\overrightarrow{W}}_R^{[\lambda ]}=\oplus {\overrightarrow{W}}_R({\overrightarrow{J}}_t,a)$ and define ${\overrightarrow{W}}^{[\lambda ]}$ to be the nilpotent 2-sided ideal ${\overrightarrow{W}}^{[\lambda ]}={\overrightarrow{W}}_L^{[\lambda ]}+{\overrightarrow{W}}_R^{[\lambda ]}$.

Definition 5.7. Define ${\overrightarrow{D}}^{[\lambda ]}$ to be the 2-sided ideal of ${\overrightarrow{I}}_f^G(x,2k)$ given by the linear span of all vectors $a\otimes b\otimes x$, where a and b are arbitrary and $x\in {\widehat{\widehat{S}}}_m^{[\lambda ]}$.

Note that ${\overrightarrow{I}}_f^G(x,2k)$ is the direct sum of the ${\overrightarrow{D}}^{[\lambda ]}$.

Proposition 5.8. ${\overrightarrow{D}}^{[\lambda ]}/{\overrightarrow{W}}^{[\lambda ]}$ is canonically isomorphic to the full matrix ring $End({\overrightarrow{R}}^{[\lambda ]})$. Recall that ${\overrightarrow{R}}^{[\lambda ]}$ is the range of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$.

Proof. Instead of m,k partial 1-factors and symmetric group, the generalised m,k signed partial 1-factors and the generalised symmetric group are used to prove the theorem. We give it here for the sake of completion.

Given eigen vectors v^(r) and v^(s) define

$$Z(v^{(r)},v^{(s)})={\left({\mu }^{(r)}{\mu }^{(s)}\right)}^{-1}{\displaystyle \sum {\left(v^{(r)}\right)}_{a,i}}{\left(v^{(s)}\right)}_{b,j}a\otimes b\otimes x_i{y}_j.$$

Taking the product of $Z(v^{(r)},v^{(s)})$ and $Z(v^{(t)},v^{(u)})$ we obtain

$$\begin{array}{l}Z(v^{(r)},v^{(s)})Z(v^{(t)},v^{(u)})\\ ={\left({\mu }^{(r)}{\mu }^{(s)}{\mu }^{(t)}{\mu }^{(u)}\right)}^{-1}\\ {\displaystyle \sum {\left(v^{(r)}\right)}_{a,i}}{\left(v^{(u)}\right)}_{d,l}a\otimes d\otimes {\left(v^{(s)}\right)}_{b,j}{\left(v^{(t)}\right)}_{c,k}\{x_i{y}_j<b,c>x_k{y}_l\}\\ ={\left({\mu }^{(r)}{\mu }^{(s)}{\mu }^{(t)}{\mu }^{(u)}\right)}^{-1}{\displaystyle \sum {\left(v^{(r)}\right)}_{a,i}}{\left(v^{(u)}\right)}_{d,l}\\ a\otimes d\otimes x_i{y}_j\left({\displaystyle \sum {\left(v^{(s)}\right)}_{b,j}}\left\{{\displaystyle \sum {\left(v^{(t)}\right)}_{c,k}}<b,c>x_k\right\}\right)y_l.\end{array}$$

Now ${\displaystyle \sum {\left(v^{(t)}\right)}_{c,k}}<b,c>x_k={\displaystyle \sum {\gamma }_r}{x}_r$ where γ_r is the (b,r) entry of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)v^{(t)}$. Since v^(t) is an eigenvector with eigenvalue ${\mu }^{(t)}$, ${\gamma }_r={\mu }^{(t)}{\left(v^{(t)}\right)}_{b,r}$. So

$$\begin{array}{l}{x}_i{y}_j\left({\displaystyle \sum {\left(v^{(s)}\right)}_{b,j}}\left\{{\displaystyle \sum {\left(v^{(t)}\right)}_{c,k}}<b,c>x_k\right\}\right)y_l\\ =x_i{y}_j\left({\displaystyle \sum {\left(v^{(s)}\right)}_{b,j}}\left\{{\displaystyle \sum {\gamma }_r}{x}_r\right\}\right)y_l\\ ={\mu }^{(t)}{\displaystyle \sum {\left(v^{(s)}\right)}_{b,j}}{\left(v^{(t)}\right)}_{b,r}\left\{x_i{y}_j{x}_r{y}_l\right\}\\ ={\mu }^{(t)}{x}_i{y}_l\left\{{\displaystyle \sum {\left(v^{(s)}\right)}_{b,j}}{\left(v^{(t)}\right)}_{b,j}\right\}\text{by}\text{equation}1.1\\ ={\mu }^{(t)}{x}_i{y}_l{\delta }_{s,t}\text{by}\text{the}\text{orthonormality}\text{of}\text{the}{v}^{(i)}.\end{array}$$

Therefore,

$$\begin{array}{l}Z(v^{(r)},v^{(s)})Z(v^{(t)},v^{(u)})={\left({\mu }^{(r)}{\mu }^{(s)}{\mu }^{(t)}{\mu }^{(u)}\right)}^{-1}{\displaystyle \sum {\left(v^{(r)}\right)}_{a,i}}{\left(v^{(u)}\right)}_{d,l}a\otimes d\otimes {\mu }^{(t)}{x}_i{y}_l{\delta }_{s,t}\\ ={\delta }_{s,t}Z(v^{(r)},v^{(u)}).\end{array}$$

Hence the subspace of ${\overrightarrow{D}}^{[\lambda ]}$ spanned by the $Z(v^{(r)},v^{(s)})$ is isomorphic to $End({\overrightarrow{R}}^{[\lambda ]})$.

The ideal ${\overrightarrow{D}}^{[\lambda ]}=V_{m,k}^G\otimes V_{m,k}^G\otimes {\widehat{\widehat{S}}}_m^{\lambda }$ is isomorphic as a vector space to $(V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})\otimes (V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})$ via the linear map f sending $(c\otimes {\overrightarrow{A}}_i)\otimes (d\otimes {\overrightarrow{A}}_j)$ to $c\otimes d\otimes x_i{y}_j$.

Writing $V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]}$ as ${\overrightarrow{N}}^{[\lambda ]}\oplus {\overrightarrow{R}}^{[\lambda ]}$ we have, from Propositions 5.4, 5.5 and 5.8, that

A. $f({\overrightarrow{N}}^{[\lambda ]}\otimes (V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})+(V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})\otimes {\overrightarrow{N}}^{[\lambda ]})$ is contained in the radical of $D_f^G\left(x\right)(2k)$

B. $f({\overrightarrow{R}}^{[\lambda ]}\otimes {\overrightarrow{R}}^{[\lambda ]})$ is a full matrix ring.

The next theorem follows immediately from A and B.

Theorem 5.9. With notation as above:

• 1. Let ${\overrightarrow{W}}^{[\lambda ]}=f\left({\overrightarrow{N}}^{[\lambda ]}\otimes (V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})+(V_{m,k}^G\otimes {\overrightarrow{S}}^{[\lambda ]})\otimes {\overrightarrow{N}}^{[\lambda ]}\right)$. Then ${\overrightarrow{W}}^{[\lambda ]}$ is the intersection of the radical of ${\overrightarrow{I}}_f^G(x,2k)$ with ${\overrightarrow{D}}^{[\lambda ]}$.

• 2. ${\overrightarrow{D}}^{[\lambda ]}/{\overrightarrow{W}}^{[\lambda ]}$ is a full matrix ring which is canonically isomorphic to $End({\overrightarrow{R}}^{[\lambda ]})$.