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The winger, 22, was dismissed in the loss to St Johnstone after appearing to react to Stephen Mitchell's award of a foul throw.
Tynecastle head coach Ian Cathro said after the game he would review the incident, but backed his player.
But BBC Scotland has learned the club is not planning to lodge an appeal.
Nicholson will be suspended for Saturday's home match against Dundee.
"I want to see the footage [of Nicholson's red card]," Cathro said in his post-match interview following Wednesday's defeat in Perth.
"I don't think there was any aggression directed in that. I take that quite clearly from how Sam describes the situation and from other players around it. That's something that needs looked at."
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With those words, marking the killing of PC Keith Palmer, deputy Speaker Lindsay Hoyle reminded us that Parliament isn't just about the politicians.
For many MPs, caught in the Commons chamber as Wednesday's attack began, their first thoughts were for their staff in other parts of the parliamentary estate.
The police put the Palace of Westminster in lockdown as firearms officers began a sweep of the area.
Plaid Cymru MP's assistant Rhian Medi Roberts told was separated from her two-year-old son Owain for "a good couple of hours" during the lockdown.
Owain was kept with other children in the parliamentary nursery - in a corridor away from windows - while Ms Roberts was told to stay in the Plaid Cymru office in the Norman Shaw buildings.
"I rushed over to the nursery as soon as we were 'freed'," said Rhian.  "But I knew him to be safe, and would be happy with his friends and carers, who were fantastic.
"Also, I knew from the constant email updates from the nursery that they fed the children a lovely dinner, which they would enjoy."
Welsh Secretary Alun Cairns had been in a meeting with the prime minister only moments before the attack. He was told to stay in his ministerial office and not to move.
Labour MP Mark Tami was locked in the Commons chamber. He said the scariest moment had been when a police marksman jumped through the door to the Members' Lobby as part of the sweep of the estate.
Liberal Democrat Lord Roberts of Llandudno said he was "old enough [81] to have had a good cry" about the attack. He was among peers allowed to visit the Commons canteen before the lockdown was over.
Others were evacuated to Westminster Abbey, where they were briefed by police. Wales Office Minister Lord Bourne of Aberystwyth tweeted: "Senseless atrocity. Unspeakable and condemned by all faiths."
On Thursday, the mood in Parliament is sombre but defiant. Welsh Liberal Democrat leader Mark Williams said: "It's an incredibly sombre, quieter and reflective place. I've just come back from the chamber where we had our daily prayers, and prayers that perhaps normally aren't exactly packed, the place was packed.
"And the mood of the house, is heartfelt sadness and extending our condolences to PC Palmer who was defending this place and defending democracy - and paid a tragic price."
Hearts will not appeal against the red card Sam Nicholson received for an alleged spit in the direction of an assistant referee at McDiarmid Park.
[NEXT_CONCEPT]
"We're in a village and our village policeman has been murdered."