MSPs have really hit the ground strolling since returning to Holyrood after the election.

Perhaps they can be excused their relaxed return to work. It was a long campaign after all, and only a year after the last long campaign. And that, you'll recall, followed a very long campaign over the independence referendum.

The start of previous parliaments inspired a mood of excitement and a rolling up of sleeves. This time, there seems to be weariness in the air.

The parties are taking a deep breath rather than sprinting ahead. There has been little appetite for accelerating the necessary housekeeping that attends the start of each new session.

Nicola Sturgeon was formally elected first minister and announced her cabinet and ministerial appointments. Holyrood's corporate body – the MSPs who act as non-executive directors of the parliament as an institution – has been selected. The business bureau, the gang of party whips who thrash out the daily running order of debates, questions and votes, is now in place.

But we're still a few days away from any actual business. We're not expecting to see the composition of the committees until Tuesday. MSPs are likely to be troubled by a debate on Thursday but the first First Minister's Question of the session has not been pencilled in until the following week. That would be a month after polling day, if you are counting.

The slow start isn't entirely down to post-election exhaustion. For the first time – and to the frustration of some of them – Holyrood's 47 new faces are undergoing induction courses, learning how to operate their electronic voting boxes work and the like.

Thought is also being given to the EU referendum campaign.

Overshadowed up to now by the Holyrood election, the June 23 poll is moving up the agenda.

Scottish Vote Leave chief Tom Harris this week urged Eurosceptic MSPs to stand up and be counted (only two of them appeared for a campaign photocall at Holyrood on Wednesday) and Jim Sillars, the former SNP deputy leader who could become one of Scotland's most effective Brexit advocates, outlined his pitch to pro-independence voters a couple of days later.

The pro-EU party machines are about to swing into action, though not before Scots LibDem leader Willie Rennie had chance to accuse the SNP of complacency.

But for all that, Scottish politics is dangling in a rare, if brief, state of limbo and, without much to go on, observers are trying to weigh up the dynamics of parliament in which the SNP is not quite as dominant as it used to be.

In his doomed but dignified bid to be elected first minister, Mr Rennie argued all parties were equal because none has a majority. That's far from the case: some are a lot more equal than others. While Ms Sturgeon will need the support of other parties on some issues, on many others she will not. Occasions when the Conservatives combine with Labour, the LibDems and Greens are likely to be few and far between.

The leaders of Labour, the LibDems and the Greens met last week to discuss how they might work together to secure their shared goal of increasing income tax in order to protect public services.

The Tories are convinced this "progressive alliance" is doomed to fail. And, while they understand Ms Sturgeon's determination to attack Ruth Davidson at every turn, they have not given up hope of influencing her government.

In a speech to MSPs this week, the new first minister reached out to her opponents on the left. But she will not be willing to give ground to them, Tory strategists believe, because the numbers she worries most about are not her 63 MSPs and the parliamentary arithmetic of winning votes but her party's slashed majorities in its heartland areas.

In Perthshire, John Swinney's majority slumped from 10,000 to 3,000 and Roseanna Cunningham's from 7,000 to less than 1,500 under pressure from Tory candidates. A similar story unfolded across the North East. With an eye on the next election, the SNP will be inclined to stick to the centre ground.

The Tories believe they can guarantee there is no drift to the left by putting forward moderate, "reasonable" alternatives.

If they are right, the parties which fought the election promising to use Holyrood's new tax powers to challenge austerity could be in for a frustrating time.