Some definitions...
Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on each monomial is $1$.
Definition 2: A polynomial $f(X_1,\dots ,X_i \dots ,X_j,\dots ,X_d)$ is $t$-alternating if the following condition holds for each $i,j$ with $1\leq i<j\leq t$: writing $X_i$ in the place of $X_j$ yields $f(\dots ,X_i \dots ,X_j,\dots )=0$.
Definition 3: a polynomial $f$ is $t$-normal if $f$ is $t$-linear and $t$-alternating.
For Rowen, given $\pi,\sigma$ in $\textrm{Sym}(n)$, the composition $\pi\sigma$ denote, first $\pi$ and then $\sigma$.
Assume throughout this text that $f(X_1,\dots ,X_d)$ is $t$-linear.
Definition 4: If $\pi\in \textrm{Sym}(k),\; k\leq t$, define $f_{(k,\pi)}$ as the sum of those monomials of $f$ in which $X_1,\dots ,X_k$ appear in the order $X_{\pi1},\dots ,X_{\pi k}$; write $f_{(k)}$ for $f_{(k,1)}$.
On the book Polynomial Identities in Ring theory, in the first chapter, section 1.2 (Facts about Normal Polynomial), Rowen proves the following proposition:
Proposition 1.2.3: A $t$-linear polynomial $f$ is $t$-normal iff $(ij)\circ f=-f$ for all $i<j\leq t$.
The implication $(\Rightarrow)$ is OK. But in the implication $(\Leftarrow)$, Rowen makes the following affirmation ''Suppose $(ij)\circ f$ for all $i<j\leq t$. Then, writing $f=\sum_{\pi \in \textrm{Sym}(t)}f_{(t,\pi)}$, we have $(ij)\circ f_{(t,\pi)}=-f_{(t,(ij)\pi)}$ for all $\pi \in \textrm{Sym}(t)$.''
I did not understand if the notation $(ij)\circ f_{(t,\pi)}$ means $[(ij)\circ f]_{(t,\pi)}$ or $(ij)\circ [f_{(t,\pi)}]$.
See it...
Suppose that $(ij)\circ f_{(t,\pi)}$ means $(ij)\circ [f_{(t,\pi)}]$. The polynomial $$f(X_1,X_2,X_3)=X_1X_2X_3 + X_2X_3X_1 + X_3X_1X_2 - X_1X_3X_2 - X_2X_1X_3 - X_3X_2X_1$$
is $3$-linear and for all $i,j$ such that $i<j\leq 3$, we have $(ij)\circ f=-j$. For $(ij)=(12)$ and $\pi=(132)$, we have $(ij)\circ f_{(t,\pi)}=(12)\circ f_{(3,\pi)}=(12)\circ [f_{(3,\pi)}]=X_3X_2X_1$ (since $[f_{(3,\pi)}]=X_3X_1X_2$).
But, since $(12)\pi=(12)(132)=(23)$, we have $-f_{(3,(12)\pi)}=-f_{(3,(23))}=-(-X_1X_3X_2)=X_1X_3X_2$. Thus, $(ij)\circ f_{(t,\pi)}\neq-f_{(t,(ij)\pi)}$. On the other hand, in this example, $[(ij)\circ f]_{(t,\pi)}=-f_{(t,(ij)\pi)}$
Now suppose that $(ij)\circ f_{(t,\pi)}$ means $[(ij)\circ f]_{(t,\pi)}$. On the same page of the book there is a theorem that says...
Theorem 1.2.5 (ii) $f$ is $t$-normal iff $f=\sum_{\pi\in\textrm{Sym}(t)}(\textrm{sg}\pi)\pi\circ f_{(t)}$.
Assuming that $(ij)\circ f_{(t,\pi)}$ means $[(ij)\circ f]_{(t,\pi)}$, this theorem does not apply to the polynomial $$P_t(X_1,\dots ,X_t)=\sum_{\pi \in \textrm{Sym}(t)}(\textrm{sg}\pi)X_{\pi1}\cdots X_{\pi t}.$$
Something is wrong, but do not know what it is. Could anyone help me with this please? What does it means $(ij)\circ f_{(t,\pi)}$?
Sorry for my english. Thank you.
