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I need to make a figure like the attached image, which contains multiple arrays, two down arrows over the specific separator lines of the first array and text above these two arrows. I could make the arrays in align environment but the problem is arrows and text above them. I used the code below to make the arrays as a figure:

beginfigure[!ht]
beginalign*
 overrightarrowP_1 
 &left[
 beginarrayc
 alpha_1 & alpha_2 & alpha_3 & alpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8 & alpha_9 \
 beta_1 & beta_2 & beta_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta_9 \
 endarray
 right] \
 overrightarrowP_2 
 &left[
 beginarrayc
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta^prime_9 \
 endarray
 right] \
 \
 overrightarrowS_1 
 &left[
 beginarrayc
 alpha_1 & alpha_2 & alpha_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha_9 \
 beta_1 & beta_2 & beta_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta_9 \
 endarray
 right] \
 overrightarrowS_2 
 &left[
 beginarrayc
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta^prime_9 \
 endarray
 right]
endalign*
captionAn example
endfigure

After the new tikzmark library, which contains the fabulous command tikzmarknode, was uploaded to CTAN and became part of the standard TeX installations, there is no longer a real need to translate existing tables or matrices to TikZ since now you can use tikzmarknode to do all the things that were harder to accomplish before. That is, you can keep what you have and just add the annotations with TikZ and with minimal effort.

documentclassarticle
usepackageamsmath
usepackagetikz
usetikzlibrarytikzmark,calc
begindocument
beginfigure[!ht]
vspace*0.6cm
beginalign*
 overrightarrowP_1 
 &left[
 beginarrayc
 alpha_1 & alpha_2 & tikzmarknodea3alpha_3 & tikzmarknodea4alpha_4 & alpha_5 & alpha_6 & alpha_7 & tikzmarknodea8alpha_8 & tikzmarknodea9alpha_9 \
 beta_1 & beta_2 & beta_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta_9 \
 endarray
 right] \
 overrightarrowP_2 
 &left[
 beginarrayc
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta^prime_9 \
 endarray
 right] \
 \
 overrightarrowS_1 
 &left[
 beginarrayc
 alpha_1 & alpha_2 & alpha_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha_9 \
 beta_1 & beta_2 & beta_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta_9 \
 endarray
 right] \
 overrightarrowS_2 
 &left[
 beginarrayc
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta^prime_9 \
 endarray
 right]
endalign*
begintikzpicture[overlay,remember picture]
draw[line width=1mm,latex-,red] ($(a3.north east)!0.5!(a4.north west)+(0,0.2)$) -- ++(0,0.6) node[black,above]$P_1$;
draw[line width=1mm,latex-,red] ($(a8.north east)!0.5!(a9.north west)+(0,0.2)$) -- ++(0,0.6)node[black,above]$P_2$;
endtikzpicture
captionAn example
endfigure
enddocument

To understand what's going on here, look e.g. at

draw[line width=1mm,latex-,red] ($(a3.north east)!0.5!(a4.north west)+(0,0.2)$) -- ++(0,0.6) node[black,above]$P_1$;

Here +(0,0.2) determines how much the tip (i.e. lower end) of the arrow gets shifted upwards, and ++(0,0.6) determines the length of the arrow. If you increase 0.2, the whole arrow will move upwards, and if you increase/decrease 0.6, the arrow will get longer/shorter.

since you include this two equations as figure, why not draw them? for example, draw them with tikz and use of the library matrix:

documentclassarticle
usepackagetikz
usetikzlibraryarrows.meta, matrix, positioning

begindocument
 beginfigure[htb]
 centering
begintikzpicture[
node distance = 9mm and 1.5mm,
 arr/.style = draw=red, very thick, Trinagle-, shorten <=1mm,
 mtrx/.style = matrix of math nodes,
 nodes=minimum width=2em,
 left delimiter=[,
 right delimiter=],
 inner ysep=1pt, inner xsep=-2pt
 ]
matrix (m) [mtrx]

alpha_1 & alpha_2 & alpha_3 & alpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8 & alpha_9 \
beta_1 & beta_2 & beta_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta_9 \
;
node[left=of m] $vecP=$;
% 
matrix (n) [mtrx, right=of n0]

alpha_1' & alpha_2' & alpha_3' & alpha_4' & alpha_5' & alpha_6' & alpha_7' & alpha_8' & alpha_9' \
beta_1' & beta_2' & beta_3' & beta_4' & beta_5' & beta_6' & beta_7' & beta_8' & beta_9' \
;
node[left=of n]$vecS_1$;
%
foreach i in 1,...,8

draw (m-1-i.north east) -- (m-2-i.south east);
draw (n-1-i.north east) -- (n-2-i.south east);

path[arr]
 (m-1-3.north east) -- + (0,0.5) node[above] $mathbfP1$;
path[arr]
 (m-1-8.north east) -- + (0,0.5) node[above] $mathbfP2$;
endtikzpicture

captionAn example of applied two-point quantum crossover
labelfig:equation
endfigure
enddocument

You can place the P_1 and P_2 markers using eso-pic's absolute positioning mechanism - placing it in the ForeGround of the current page (via AddToShipoutPictureFG*). zref's savepos module allows you to capture the x,y coordinates of a <label> (set using zsavepos<label>) which can be retrieved using zposx/zposy (both expandable).

documentclassarticle

usepackageamsmath,zref-savepos,eso-pic

begindocument

beginalign*
 & \ % Just to leave an empty line
 overrightarrowP_1 
 & left[
 beginarray *8c c 
 alpha_1 & alpha_2 & alpha_3zsaveposP1L & zsaveposP1Ralpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8zsaveposP2L & zsaveposP2Ralpha_9 \
 beta_1 & beta_2 & beta_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta_9 \
 endarray
 right] \
 overrightarrowP_2 
 & left[
 beginarray *8c c 
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta^prime_9 \
 endarray
 right] \
 \
 overrightarrowS_1 
 & left[
 beginarray *8c c 
 alpha_1 & alpha_2 & alpha_3 & alpha^prime_4 & alpha^prime_5 & alpha^prime_6 & alpha^prime_7 & alpha^prime_8 & alpha_9 \
 beta_1 & beta_2 & beta_3 & beta^prime_4 & beta^prime_5 & beta^prime_6 & beta^prime_7 & beta^prime_8 & beta_9 \
 endarray
 right] \
 overrightarrowS_2 
 & left[
 beginarray *8c c 
 alpha^prime_1 & alpha^prime_2 & alpha^prime_3 & alpha_4 & alpha_5 & alpha_6 & alpha_7 & alpha_8 & alpha^prime_9 \
 beta^prime_1 & beta^prime_2 & beta^prime_3 & beta_4 & beta_5 & beta_6 & beta_7 & beta_8 & beta^prime_9 \
 endarray
 right]
endalign*
AddToShipoutPictureFG*%
 AtPageLowerLeft%
 hspace0.5dimexprzposxP1Lsp+zposxP1Rsprelax%
 raiseboxdimexprzposyP1Lsp+1.5baselineskipmakebox[0pt]$substackP_1 \ downarrow$%
 %
 AtPageLowerLeft%
 hspace0.5dimexprzposxP2Lsp+zposxP2Rsprelax%
 raiseboxdimexprzposyP1Lsp+1.5baselineskipmakebox[0pt]$substackP_2 \ downarrow$%
 %


enddocument

Above I place two markers around the array column divider for each of P_1 and P_2 (denoted with a Left and Right suffix) and position the labels to be at the horizontal centre of these markers.

An empty line within the align* is left at the top in order to avoid unnecessary interaction with content appearing above the environment.

Since the markers use TeX's label-ref-like system, any change in the location of the markers requires an additional compilation so reference can settle.