Fulton's *Young tableaux* is one of the best texts on the subject, one which I
often recommend and cite for reference. Unlike Fulton/Lang and
Fulton/Harris,
it is neither an early-dawn draft nor a chatty informal introduction but aims
(and often succeeds) at giving a streamlined and readable presentation of the
subject. All the more I find it important to spot and correct the occasional
errors in it.

I used to think it had none, until I learned of three seriously broken proofs in rapid succession:

**Serious errors:**

**page 97, proof of Lemma 5 (§ 7.4):**The number \begin{align*} n_{\lambda} & =\left\vert \left\{ \left( p_{1},q_{1},p_{2},q_{2}\right) \ :\ p_{i}\in R\left( T\right) ,\ q_{i}\in C\left( T\right) \right. \right. \\ & \ \ \ \ \ \ \ \ \ \ \left. \left. p_{1}q_{1}p_{2}q_{2}=1, \ \operatorname*{sgn}\left( q_{1}\right) =\operatorname*{sgn}\left( q_{2}\right) \right\} \right\vert \end{align*} (where $R\left( T\right) $ is the row-fixing group and $C\left( T\right) $ the column-fixing group of a given standard tableau $T$) is wrong, as it fails to satisfy the equation $\left( b_{T}\cdot a_{T}\right) \cdot v_{T} =n_{\lambda}\cdot v_{T}$. (It would satisfy the equation if it was true that any permutations $p_{1},p_{2}\in R\left( T\right) $ and $q_{1},q_{2}\in C\left( T\right) $ satisfying $p_{1}q_{1}p_{2}q_{2}=1$ would satisfy $\operatorname*{sgn}\left( q_{1}\right) =\operatorname*{sgn}\left( q_{2}\right) $. But this fails when $n=8$ and $\lambda = \left( 3,3,2\right) $ and $T= \begin{array} [c]{ccc} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & \end{array} $ and $p_{1}=\left[ 23164578\right] $ and $q_{1}=\left[ 48372615\right] $ and $p_{2}=\left[ 32146587\right] $ and $q_{2}=\left[ 18675342\right] $, where all permutations are given in one-line notation. Maybe it holds for two-row partitions? I don't know.)

Unfortunately, this is not easy to fix. We can make the equations $\left(
b_{T}\cdot a_{T}\right) \cdot v_{T}=n_{\lambda}\cdot v_{T}$ and $\left(
a_{T}\cdot b_{T}\right) \cdot\widetilde{v}_{T}=n_{\lambda}\cdot
\widetilde{v}_{T}$ true if we replace the number $n_{\lambda}$ by
\begin{align*}
n_{\lambda}^{\prime}:=\sum_{\substack{\left( p_{1},q_{1},p_{2},q_{2}\right)
;\\p_{i}\in R\left( T\right) ,\ q_{i}\in C\left( T\right) ;\\
p_1q_{1}p_{2}q_{2}=1}}\operatorname*{sgn}\left( q_{1}\right) \cdot
\operatorname*{sgn}\left( q_{2}\right) .
\end{align*}
But we also need to show that this number $n_{\lambda}^{\prime}$ is positive,
and this is not as obvious as it was for $n_{\lambda}$. Fortunately, this can
still be shown using some easily accessible results: The number $n_{\lambda
}^{\prime}$ is easily seen to be coefficient of the identity permutation $1$
in $\left( a_{T}b_{T}\right) ^{2}=c_{T}^{2}$, where $c_T := a_T b_T$ (this differs a bit from Fulton's $c_T$, which is $b_T a_T$, but this is not a major distinction). However, it is known (see,
e.g., Lemma 5.13.3 in Pavel Etingof et al., *Introduction to representation
theory*, AMS 2011) that
$c_{T}^{2}=\dfrac{n!}{\left\vert P_{\lambda}\right\vert \left\vert Q_{\lambda
}\right\vert \dim V_{\lambda}}c_{T}$ for a certain positive number $\dfrac
{n!}{\left\vert P_{\lambda}\right\vert \left\vert Q_{\lambda}\right\vert \dim
V_{\lambda}}$ (see loc. cit. for its definition). (Note that the lemma is
stated not for an arbitrary numbering $T$ but only for the specific numbering
$T_{\lambda}$ which has the numbers $1,2,\ldots,n$ filled in from top to
bottom, row by row. But this is fine, since the numbering can be chosen
arbitrarily.) Furthermore, it is easy to see that the coefficient of the
permutation $1$ in $c_{T}$ is $1$. Hence, the coefficient of $1$ in
$c_{T}^{2}=\dfrac{n!}{\left\vert P_{\lambda}\right\vert \left\vert Q_{\lambda
}\right\vert \dim V_{\lambda}}c_{T}$ is $\dfrac{n!}{\left\vert P_{\lambda
}\right\vert \left\vert Q_{\lambda}\right\vert \dim V_{\lambda}}$, which is
positive. Thus, $n_{\lambda}^{\prime}$ is positive, qed.

**page 108, proof of Lemma 2 (§ 8.1):**I am not convinced that "In the latter case, fixing $v_{p}=w_{1}$, it suffices to show that the difference of the two sides is an alternating function of $v_{1},v_{2},\ldots,v_{p},w_{2}$". The problem is that the left hand side is not linear in $v_{p}=w_{1}$, since $v_{p}$ appears twice in it (once as $v_{p}$ and once again as $w_{1}$). I don't know how easy this is to correct, but there are other proofs of Lemma 2 around in the literature (e.g., it is Exercise 6.66 in arXiv:2008.09862v3, where it is proved just using multi-column Laplace expansion).**page 113, proof of Lemma 4 (§ 8.2):**The claim that "$e_{T^{\prime}}$ occurs in $g\cdot e_{T}$ with coefficient $1$" is not obvious. More importantly, this argument only rules out other highest weight vectors of the form $e_{T}$, but not highest weight vectors that are arbitrary linear combinations of different $e_{T}$'s. I don't know if this gap can be easily fixed. See an answer below for a correct (I hope...) proof of Lemma 4.**page 118, proof of Proposition 1 (§ 8.3):**This proof makes a leap of faith in the last paragraph: How exactly does the construction of the map from $E^{\lambda}$ to $E\left( S^{\lambda}\right) $ yield that $E^{\lambda}\cong E\left( \widetilde{M}^{\lambda}\right) /E\left( Q^{\lambda}\right) $ ? There is no direct way to obtain this from the map itself, since it is not obvious that the map is injective. Instead, I suggest the following argument:

Proposition 4 in § 7.4 yields that $S^{\lambda}\cong\widetilde{M}^{\lambda }/Q^{\lambda}$, where $Q^{\lambda}$ is as defined in § 7.4. On the other hand, set $\mu=\widetilde{\lambda}$ and $\ell=\lambda_{1}$. Then, (11) shows that there is an isomorphism \begin{align*} H:E\left( \widetilde{M}^{\lambda}\right) \rightarrow\wedge^{\mu_{1}}\left( E\right) \otimes\cdots\otimes\wedge^{\mu_{\ell}}\left( E\right) \end{align*} (which sends any pure tensor $\left( v_{1}\otimes v_{2}\otimes\cdots\otimes v_{n}\right) \otimes_{\mathbb{C}\left[ S_{n}\right] }\left[ T\right] \in E\left( M^{\lambda}\right) $ to \begin{align*} \left( v_{T\left( 1,1\right) }\wedge v_{T\left( 1,2\right) }\wedge \cdots\wedge v_{T\left( 1,\mu_{1}\right) }\right) \otimes\cdots \otimes\left( v_{T\left( \ell,1\right) }\wedge v_{T\left( \ell,2\right) }\wedge\cdots\wedge v_{T\left( \ell,\mu_{\ell}\right) }\right) \end{align*} ).

However, the inclusion $Q^{\lambda}\rightarrow\widetilde{M}^{\lambda}$ yields (by the functoriality of $E$) a map from $E\left( Q^{\lambda}\right) $ to $E\left( \widetilde{M}^{\lambda}\right) $. Due to how $Q^{\lambda}$ was defined, the image of this map is the span of all the differences $\mathbf{v}\left( T\right) \otimes v_{T}-\sum_{S}\mathbf{v}\left( S\right) \otimes v_{S}$, where $\mathbf{v}$ ranges over all pure tensors in $E^{\otimes n}$, where $T$ varies over all numberings of $\lambda$, where $j$ and $k$ vary over all integers satisfying $1\leq j\leq\ell-1$ and $1\leq k\leq\mu_{j+1}$, and where the sum is over all numberings $S$ in $\pi_{j,k}\left( T\right) $. Under our above isomorphism $H$, these differences $\mathbf{v}\left( T\right) \otimes v_{T}-\sum_{S}\mathbf{v}\left( S\right) \otimes v_{S}$ turn into exactly the generators $\wedge\mathbf{v}-\sum\wedge\mathbf{w}$ of the subspace $Q^{\lambda}\left( E\right) $ defined in § 8.1. Hence, under the isomorphism $H$, the image of $E\left( Q^{\lambda}\right) $ in $E\left( \widetilde{M}^{\lambda}\right) $ becomes the subspace $Q^{\lambda}\left( E\right) $. Therefore, \begin{align*} E\left( \widetilde{M}^{\lambda}\right) /E\left( Q^{\lambda}\right) \cong\left( \wedge^{\mu_{1}}\left( E\right) \otimes\cdots\otimes\wedge ^{\mu_{\ell}}\left( E\right) \right) /Q^{\lambda}\left( E\right) =E^{\lambda} \end{align*} (by (4)).

But the functor $E$ is right-exact (since it is a tensor functor), and thus we have $E\left( \widetilde{M}^{\lambda}/Q^{\lambda}\right) \cong E\left( \widetilde{M}^{\lambda}\right) /E\left( Q^{\lambda}\right) \cong E^{\lambda}$. In other words, $E\left( S^{\lambda}\right) \cong E^{\lambda}$ (since $S^{\lambda}\cong\widetilde{M}^{\lambda}/Q^{\lambda}$). This proves Proposition 1.

**Typos:**

**page 73, proof of Proposition 1 (§ 6.1):**In the last identity, replace $\prod_{i=1}^m\left(1-x_jt\right)$ by $\prod_{i=1}^m\left(1-x_it\right)$.**page 91, proof of Theorem (§ 7.3):**On line 4 of the proof, replace $M^{\left(\lambda\right)}$ by $M^\lambda$.**page 94, proof of Proposition 3 (§ 7.3):**Three missing $)$ parentheses.**page 94, Exercise 11 (§ 7.3):**Replace "$\mathbb{C}_{n}$" by "$\mathbb{C}^{n}$".**page 95, middle of the page (§ 7.4):**"subspace generated by all $\left[T\right] - \operatorname{sgn}\left(q\right)\left[T\right]$" should be "subspace generated by all $\left[qT\right] - \operatorname{sgn}\left(q\right)\left[T\right]$".**page 101, proof of Proposition 4 (§ 7.4):**The last sum ($\sum_{\ell=0}^k \left(-1\right)^\ell \dbinom{k}{\ell}$) should actually be $\sum_{\ell=0}^{k-m} \left(-1\right)^\ell \dbinom{k-m}{\ell}$. (Of course, this still equals $0$.)**page 103, Exercise 21 (§ 7.4):**The sum $\sum_{\lambda, \nu}$ should actually be a sum $\sum_{\lambda, \mu}$.**page 111, Exercise 2 (§ 8.1):**Of course, in (iii), the partition $\lambda$ should be assumed to be nonempty.**page 111, paragraph between Exercises 2 and 3 (§ 8.1):**"a left action of the algebra $\operatorname{End}_R\left(E\right)$ on $E^\lambda$" should be "a left action of the multiplicative monoid $\operatorname{End}_R\left(E\right)$ on $E^\lambda$" (since this action is not linear in the acting element: i.e., two endomorphisms $U, V \in \operatorname{End}_R\left(E\right)$ will rarely satisfy $\left(U+V\right)\mathbf{e} = U\mathbf{e} + V\mathbf{e}$ for all $\mathbf{e} \in E^\lambda$). Likewise, the "$M_mR$" has to be understood as a multiplicative monoid, not as an algebra. Similarly, just below Exercise 3, "The algebra $M_mR$ also acts on the left on the $R$-algebra $R\left[Z\right]$" should be read as "The monoid $M_mR$ also acts by $R$-algebra endomorphisms on the left on the $R$-algebra $R\left[Z\right]$".**page 114, middle of the page (§ 8.2):**"if and only if $\lambda _{i}+k=\lambda_{i}+k^{\prime}$ for all $i$" should be "if and only if $\lambda_{i}+k=\lambda_{i}^{\prime}+k^{\prime}$ for all $i$".**page 118, proof of Proposition 1 (§ 8.3):**The $\lambda$ in "$E\left(S_\lambda\right)$" should be a superscript, not a subscript.**page 123, line -8 (§ 8.3):**Replace "presentation" by "representation".**page 125, generators of $Q$ (§ 8.4):**Both $w_p$s in the formula should be $w_q$s.**page 133, Exercise 1 (§ 9.1):**The $i_{d+1}$ subscript should be $j_{d+1}$.**page 133, proof of Lemma 1 (§ 9.1):**"Fix some $i_i, \ldots, i_d$" should be "Fix some $i_1, \ldots, i_d$", of course.

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